Differential Equations With Variable Separable
Class 12th Mathematics RS Aggarwal Solution
- {dy}/{dx} = ( 1+x^{2} ) ( 1+y^{2} ) Find the general solution of each of the…
- x^{4} {dy}/{dx} = - y^{4} Find the general solution of each of the…
- {dy}/{dx} = 1+x+y+xy Find the general solution of each of the following…
- {dy}/{dx} = 1-x+y-xy Find the general solution of each of the following…
- (x-1) {dy}/{dx} = 2x^{3}y Find the general solution of each of the following…
- {dy}/{dx} = e^{x+y} Find the general solution of each of the following…
- ( e^{x} + e^{-x} ) dy - ( e^{x} - e^{-x} ) dx = 0 Find the general solution of…
- Find the general solution of each of the following differential equations:…
- e^{2x-3y} dx+e^{2y-3x}dy = 0 Find the general solution of each of the following…
- ex tan y dx + (1 – ex) sec2 y dy = 0 Find the general solution of each of the…
- sec2x tan y dx + sec2y tan x dy = 0 Find the general solution of each of the…
- cos x(1 + cos y)dx – sin y(1 + sin x)dy = 0 Find the general solution of each…
- . cos ( {dy}/{dx} ) = a . where a ∈ R and y = 2 when x = 0. For each of the…
- {dy}/{dx} = - 4xy^{2} it being given that y = 1 when x = 0. For each of the…
- x dy = (2x2 + 1) dx (x ≠ 0), given thaty = 1 when x = 1. For each of the…
- {dy}/{dx} = ytanx it being given that y = 1 when x = 0. For each of the…
- {dy}/{dx} = frac {x-1}/{y+2} Find the general solution of each of the following…
- {dy}/{dx} = frac {x}/{ ( x^{2} + 1 ) } Find the general solution of each of the…
- {dy}/{dx} = (1+x) ( 1+y^{2} ) Find the general solution of each of the…
- ( 1+x^{2} ) {dy}/{dx} = xy Find the general solution of each of the following…
- {dy}/{dx} + y = 1 ( y not equal 1 ) Find the general solution of each of the…
- {dy}/{dx} + root { frac { 1-y^{2} }/{ 1-x^{2} } } = 0 Find the general…
- x {dy}/{dx} + y = y^{2} Find the general solution of each of the following…
- x2 (y + 1) dx + y2 (x – 1) dy = 0 Find the general solution of each of the…
- y ( 1-x^{2} ) {dy}/{dx} = x ( 1+y^{2} ) Find the general solution of each of…
- y log y dx – x dy = 0 Find the general solution of each of the following…
- x(x2 – x2 y2) dy + y(y2 + x2y2) dx = 0 Find the general solution of each of…
- (1 – x2) dy + xy (1 – y) dx = 0 Find the general solution of each of the…
- (1 – x2)(1 – y) dx = xy (1 + y) dy Find the general solution of each of the…
- (y + xy) dx + (x – xy2) dy = 0 Find the general solution of each of the…
- (x2 – yx2) dy + (y2 + xy2) dx = 0 Find the general solution of each of the…
- ( x^{2}y-x^{2} ) dx + ( xy^{2} - y^{2} ) dy = 0 Find the general solution of…
- x root { 1+y^{2} } dx+y sqrt { 1+x^{2} } dy = 0 Find the general solution of…
- {dy}/{dx} = e^{x+y} + x^{2}e^{y} Find the general solution of each of the…
- {dy}/{dx} = frac { 3e^{2x} + 3d^{4x} }/{ e^{x} + e^{-x} } Find the general…
- 3e^{x}tanydx + ( 1-e^{x} ) sec^{2}ydy = 0 Find the general solution of each of…
- e^{y} ( 1+x^{2} ) dy - {x}/{y} dx = 0 Find the general solution of each of…
- {dy}/{dx} = e^{x+y} + e^{x-y} Find the general solution of each of the…
- (ey + 1) cos x dx + ey sin x dy = 0 Find the general solution of each of the…
- {dy}/{dx} + frac {xy+y}/{xy+x} = 0 Find the general solution of each of the…
- root { 1-x^{4} } dy = xdx Find the general solution of each of the following…
- cosecxlogy {dy}/{dx} + x^{2}y = 0 Find the general solution of each of the…
- ydx + ( 1+x^{2} ) tan^{-1}xdy = 0 Find the general solution of each of the…
- {1}/{x} c. frac {dy}/{dx} = tan^{-1}x Find the general solution of each of…
- e^{x}root { 1-y^{2} } dx + {y}/{x} dy = 0 Find the general solution of each…
- {dy}/{dx} = frac {1-cosx}/{1+cosx} Find the general solution of each of the…
- (cosx) {dy}/{dx} + cos2x = cos3x Find the general solution of each of the…
- {dy}/{dx} + frac { (1+cos2y) }/{ (1-cos2x) } = 0 Find the general solution of…
- {dy}/{dx} + frac {cosxsiny}/{cosy} = 0 Find the general solution of each of…
- cos x(1 + cos y)dx – sin y(1 + sin x)dy = 0 Find the general solution of each…
- sin3 x dx – sin y dy = 0 Find the general solution of each of the following…
- {dy}/{dx} + sin (x+y) = sin (x-y) Find the general solution of each of the…
- {1}/{x} cos^{2}ydy + frac {1}/{y} cos^{2}xdx = 0 Find the general solution of…
- {dy}/{dx} = sin^{3}xcos^{2}x+xe^{x} Find the general solution of each of the…
- Find the particular solution of the differential equation {dy}/{dx} =…
- Find the particular solution of the differential equation x(1 + y2) dx-y(1 +…
- Find the particular solution of the differential equation log ( {dy}/{dx} )…
- Solve the differential equation (x2 – yx2)dy + (y2 + x2y2) dx = 0, given that…
- Find the particular solution of the differential equation e^{x}root {…
- Find the particular solution of the differential equation {dy}/{dx} = frac…
- Solve the differential equation {dy}/{dx} = ysin2x given that y(0) = 1.…
- Solve the differential equation (x+1) {dy}/{dx} = 2xy given that y(2) =…
- Solve {dy}/{dx} = x (2logx+1) given that y = 0 when x = 2.
- Solve ( x^{3} + x^{2} + x+1 ) {dy}/{dx} = 2x^{2} + x given that y = 1…
- Solve {dy}/{dx} = ytanx given that y = 1 when x = 0.
- Solve {dy}/{dx} = y^{2}tan2x given that y = 2 when x = 0.
- Solve {dy}/{dx} = ycot2x given that y = 2 when x = { pi }/{4}…
- Solve (1 + x2) sec2 y dy + 2x tan y dx = 0, given that y = { pi }/{4}…
- Find the equation of the curve passing through the point ( 0 , { pi }/{4}…
- Find the equation of a curve which passes through the origin and whose…
- A curve passes through the point (0, -2) and at any point (x, y) of the curve,…
- A curve passes through the point (-1, 1) and at any point (x, y) of the curve,…
- In a bank, principal increases at the rate fo r% per annum. Find the value of…
- In a bank, principal increases at the rate of 5% per annum. An amount of `…
- The volume of a spherical balloon being inflated changes at a constant rate.…
- In a culture the bacteria count is 100000. The number is increased by 10% in 2…
Exercise 19a
Question 1.Find the general solution of each of the following differential equations:
Answer:
Rearranging the terms,we get:
Integrating both the sides we get,
…(
)
Ans:
Question 2.
Find the general solution of each of the following differential equations:
Answer:
Integrating both the sides we get,
…(3c’ = c)
Question 3.
Find the general solution of each of the following differential equations:
Answer:
Rearranging the terms we get:
Integrating both the sides we get,
…(
Ans:
Question 4.
Find the general solution of each of the following differential equations:
Answer:
Rearranging the terms we get:
Integrating both the sides we get,
…(
Ans:
Question 5.
Find the general solution of each of the following differential equations:
Answer:
Separating the variables we get:
Integrating both the sides we get,
Ans:
Question 6.
Find the general solution of each of the following differential equations:
Answer:
Rearringing the terms we get:
Integrating both the sides we get,
Ans:ex + e - y = c
Question 7.
Find the general solution of each of the following differential equations:
Answer:
(ex + e - x )dy - (ex - e - x)dx = 0
Integrating both the sides we get,
y = log|ex + e - x| + c …(
Ans:y = log|ex + e - x| + c
Question 8.
Find the general solution of each of the following differential equations:
Answer:
Given:
Integrating both the sides we get:
Ans:
Question 9.
Find the general solution of each of the following differential equations:
Answer:
e2xe - 3ydx + e2ye - 3xdy = 0
Rearringing the terms we get:
e2x + 3xdx = - e2y + 3ydy
⇒e5xdx = - e5ydy
Integrating both the sides we get:
⇒e5x + e5y = 5c’ = c
Ans: e5x + e5y = c
Question 10.
Find the general solution of each of the following differential equations:
ex tan y dx + (1 – ex) sec2 y dy = 0
Answer:
Rearranging all the terms we get:
Integrating both the sides we get:
⇒log|1 - ex| = log|tany| - logc
⇒log|1 - ex| + logc = log|tany|
⇒tany = c(1 - ex)
Ans: tany = c(1 - ex)
Question 11.
Find the general solution of each of the following differential equations:
sec2x tan y dx + sec2y tan x dy = 0
Answer:
Rearranging the terms we get:
Integrating both the sides we get:
⇒log|tanx| = - log|tany| + logc
⇒ log|tanx| + log|tany| = logc
⇒tanx.tany = c
Ans: tanx.tany = c
Question 12.
Find the general solution of each of the following differential equations:
cos x(1 + cos y)dx – sin y(1 + sin x)dy = 0
Answer:
Rearranging the terms we get:
Integrating both the sides we get:
⇒log|1 + sinx| = - log|1 + cosy| + logc
⇒log|1 + sinx| + log|1 + cosy| = logc
⇒(1 + sinx)(1 + cosy) = c
Ans: (1 + sinx)(1 + cosy) = c
Question 13.
For each of the following differential equations, find a particular solution satisfying the given condition :
.
. where a ∈ R and y = 2 when x = 0.
Answer:
⇒
⇒dy = cos - 1a dx
Integrating both the sides we get:
⇒y = xcos - 1a + c
when x = 0, y = 2
∴2 = 0 + c
∴c = 2
∴y = xcos - 1a + 2
Ans:
Question 14.
For each of the following differential equations, find a particular solution satisfying the given condition :
it being given that y = 1 when x = 0.
Answer:
Rearranging the terms we get:
Integrating both the sides we get:
⇒y - 1 = 2x2 + c
y = 1 when x = 0
⇒ (1) - 1 = 2(0)2 + c
⇒c = 1
Ans:
Question 15.
For each of the following differential equations, find a particular solution satisfying the given condition :
x dy = (2x2 + 1) dx (x ≠ 0), given that
y = 1 when x = 1.
Answer:
Rearranging the terms we get:
Integrating both the sides we get:
⇒y = x2 + log|x| + c
y = 1 when x = 1
∴1 = 12 + log1 + c
∴1 - 1 = 0 + c …(log1 = 0)
⇒c = 0
∴y = x2 + log|x|
Ans: y = x2 + log|x|
Question 16.
For each of the following differential equations, find a particular solution satisfying the given condition :
it being given that y = 1 when x = 0.
Answer:
Rearranging the terms we get:
⇒log|y| = log|secx| + logc
⇒log|y| - log|secx| = logc
⇒log|y| + log|cosx| = logc
⇒ycosx = c
y = 1 when x = 0
∴1×cos0 = c
∴c = 1
⇒ycosx = 1
⇒y = 1/cosx
⇒y = secx
Ans: y = secx
Exercise 19b
Question 1.Find the general solution of each of the following differential equations:
Answer:
(y + 2)dy = (x - 1)dx
Integrating on both sides,
Question 2.
Find the general solution of each of the following differential equations:
Answer:
Multiply and divide 2 in numerator and denominator of RHS,
Integrating on both sides
Question 3.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both sides
Question 4.
Find the general solution of each of the following differential equations:
Answer:
Multiply and divide 2 in numerator and denominator of RHS,
Integrating on both sides
⇒
1
Question 5.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both sides
Question 6.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both sides
Question 7.
Find the general solution of each of the following differential equations:
Answer:
⟹
Integrating on both the sides,
LHS:
Let 
Comparing coefficients in both the sides,
A = - 1, B = 1
RHS:
Therefore the solution of the given differential equation is
Question 8.
Find the general solution of each of the following differential equations:
x2 (y + 1) dx + y2 (x – 1) dy = 0
Answer:
Add and subtract 1 in numerators of both LHS and RHS,
By the identity,
Splitting the terms,
Integrating,
Question 9.
Find the general solution of each of the following differential equations:
Answer:
Multiply 2 in both LHS and RHS,
Integrating on both the sides,
Question 10.
Find the general solution of each of the following differential equations:
y log y dx – x dy = 0
Answer:
Integrating on both the sides,
LHS:
RHS:
Let 
So, 
Therefore the solution of the given differential equation is
Question 11.
Find the general solution of each of the following differential equations:
x(x2 – x2 y2) dy + y(y2 + x2y2) dx = 0
Answer:
Integrating ,
Question 12.
Find the general solution of each of the following differential equations:
(1 – x2) dy + xy (1 – y) dx = 0
Answer:
Integrating on both the sides,
LHS:
Let
⇒
Comparing coefficients in both the sides,
A = - 1, B = 1
RHS:
Multiply and divide 2
Therefore the solution of the given differential equation is
- 
= 
Question 13.
Find the general solution of each of the following differential equations:
(1 – x2)(1 – y) dx = xy (1 + y) dy
Answer:
Integrating on both the sides,
LHS:
RHS:
Add and subtract 1 in numerators of both LHS and RHS,
By the identity,
Splitting the terms,
Integrating,
Therefore the solution of the given differential equation is
Question 14.
Find the general solution of each of the following differential equations:
(y + xy) dx + (x – xy2) dy = 0
Answer:
Integrating ,
Question 15.
Find the general solution of each of the following differential equations:
(x2 – yx2) dy + (y2 + xy2) dx = 0
Answer:
Integrating,
Question 16.
Find the general solution of each of the following differential equations:
Answer:
Add and subtract 1 in numerators ,
By the identity,
Splitting the terms,
Integrating,
Question 17.
Find the general solution of each of the following differential equations:
Answer:
Integrating,
Question 18.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both the sides,
Question 19.
Find the general solution of each of the following differential equations:
Answer:
Considering ‘d’ as exponential ’e’
Integrating on both the sides,
Question 20.
Find the general solution of each of the following differential equations:
Answer:
⇒
Integrating on both the sides,
formula:
Question 21.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both the sides,
LHS:
By ILATE rule,
RHS:
Multiply and divide by 2
Therefore the solution of the given differential equation is
Question 22.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both the sides,
formula: 
= 
Question 23.
Find the general solution of each of the following differential equations:
(ey + 1) cos x dx + ey sin x dy = 0
Answer:
Integrating,
Question 24.
Find the general solution of each of the following differential equations:
Answer:
Integrating ,
Question 25.
Find the general solution of each of the following differential equations:
Answer:
Multiply and divide by 2,
Integrating on both the sides,
formula: 
⇒
Question 26.
Find the general solution of each of the following differential equations:
Answer:
Integrating ,
Consider the integral 
Let
So,
Consider the integral 
By ILATE rule,
Again by ILATE rule,
Therefore the solution of the given differential equation is,
Question 27.
Find the general solution of each of the following differential equations:
Answer:
Integrating,
Consider the integral 
Let 
So, 
Consider the integral 
Therefore the solution of the differential equation is
Question 28.
Find the general solution of each of the following differential equations:
Answer:
Integrating on both the sides,
(adding and subtracting 1)
Question 29.
Find the general solution of each of the following differential equations:
Answer:
Integrating,
Consider the integral 
By ILATE rule,
Consider the integral 
Its value is - 
as 
Therefore the solution of the given differential equation is
- 
Question 30.
Find the general solution of each of the following differential equations:
Answer:
can be written as 
Integrating on both the sides,
formula: 
formula:
Question 31.
Find the general solution of each of the following differential equations:
Answer:
Given: 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 32.
Find the general solution of each of the following differential equations:
Answer:
Given:
⇒ 
⇒ 
⇒ 
⇒ 
Question 33.
Find the general solution of each of the following differential equations:
Answer:
Given:
⇒ 
⇒ 
⇒ 
Question 34.
Find the general solution of each of the following differential equations:
cos x(1 + cos y)dx – sin y(1 + sin x)dy = 0
Answer:
Given: cosx(1+cosy)dx-siny(1+sinx)dy=0
Dividing the whole equation by (1+sinx)(1+cosy), we get,
⇒ 
⇒ log|1+sinx|+log|1+cosy|=logc
⇒ (1+sinx)(1+cosy)=c
Question 35.
Find the general solution of each of the following differential equations:
sin3 x dx – sin y dy = 0
Answer:
We have,
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 36.
Find the general solution of each of the following differential equations:
Answer:
⇒ 
⇒ 
(Using sin(A+B)-sin(A-B)=2sinBcosA)
⇒ 
⇒ 
⇒ 
⇒ sinx+log|cosecy-coty|+c=0
Question 37.
Find the general solution of each of the following differential equations:
Answer:
Given: 
⇒ 
⇒ 
(Using, 2cos2a=1+cos2a)
⇒ 
⇒ 
⇒ 
Question 38.
Find the general solution of each of the following differential equations:
Answer:
Here we have, 
⇒ 
Taking cosx as t we have,
⇒ 
⇒ 
So we have,
⇒ 
⇒ 
⇒ 
Question 39.
Find the particular solution of the differential equation 
given that y = 0 when x = 1.
Answer:
Given:
⇒ 
⇒ 
⇒ 
We have,
⇒ 
⇒ 
⇒ 
Question 40.
Find the particular solution of the differential equation x(1 + y2) dx
-y(1 + x2) dy = 0, given that y = 1 when x = 0.
Answer:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 41.
Find the particular solution of the differential equation 
given that y = 0 when x = 0.
Answer:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ For 
, we have
⇒ 
⇒ 
⇒ 
⇒ 
Question 42.
Solve the differential equation (x2 – yx2)
dy + (y2 + x2y2) dx = 0, given that y = 1 when x = 1.
Answer:
⇒ 
For y=1,x=1, we have,
⇒ 
⇒ 
1
Hence, the required solution is:
⇒ 
Question 43.
Find the particular solution of the differential equation 
given that y = 1 when x = 0.
Answer:
Given:
Separating the variables we get,
⇒ 
⇒ 
Substituting 
, we have,
⇒ 
For y=1 and x=0, we have,
⇒ 
⇒ 
⇒ Hence, the particular solution will be:-
⇒ 
Question 44.
Find the particular solution of the differential equation 
given that 
when x = 1.
Answer:
Given: 
⇒ 
Let 
Then,
And 
We have,
⇒ 
For 
we have,
⇒ 
Question 45.
Solve the differential equation 
given that y(0) = 1.
Answer:
We have,
⇒ 
⇒ 
For y=1, x=0, we have,
c = 
⇒ 
⇒ 
Thus,
The particular solution is:
Question 46.
Solve the differential equation 
given that y(2) = 3.
Answer:
Given: 
⇒ 
⇒ 
⇒ 
For x=2 and y=3, we have,
c = 3log3 – 4
Hence, the particular solution is,
⇒ y(x + 1)2 = 27
Question 47.
Solve 
given that y = 0 when x = 2.
Answer:
we have, 
, Integrating we get,
,
given that y=0 when x=2
⇒ 
now putting x=2 and y=0,
⇒ 
⇒ 
Thus, the solution is:
Question 48.
Solve 
given that y = 1 when x = 0.
Answer:
we have,
Given that: y=1 when x=0,
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
For y=1, when x=0, we have,
⇒ 
Question 49.
Solve given that y = 1 when x = 0.
Answer:
we have,
given that: y=1 when x=0
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
is the particular solution…
Question 50.
Solve 
given that y = 2 when x = 0.
Answer:
we have:
Given that, y=2 when x=0
⇒ 
⇒ 
…integrating both sides
⇒ 
⇒ 
⇒ 
⇒ 
…is the particular solution
Question 51.
Solve 
given that y = 2 when 
Answer:
we have
Given that, y=2 when x=
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
+c
⇒ Thus, 
The particular solution is :- 
Question 52.
Solve (1 + x2) sec2 y dy + 2x tan y dx = 0, given that 
when x = 1.
Answer:
we have, (1 + x2) sec2 y dy + 2x tan y dx = 0,
Given that, when x=1
⇒ 
⇒ 
⇒ 
⇒ 
For 
We have, 
c = 2,
Hence the required particular solution is:-
Question 53.
Find the equation of the curve passing through the point 
whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Answer:
we have, sin x cos y dx + cos x sin y dy = 0
⇒ 
⇒ 
⇒ 
⇒ 
Given that, coordinates of point
⇒ 
⇒ 
…is the required particular solution
Question 54.
Find the equation of a curve which passes through the origin and whose differential equation is 
Answer:
Given, 
⇒ 
Let 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
For the curve passes through (0,0)
We have, c =
Question 55.
A curve passes through the point (0, -2) and at any point (x, y) of the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point. Find the equation of the curve.
Answer:
Given that the product of slope of tangent and y coordinate equals the x-coordinate i.e., 
We have, 
⇒ 
⇒ 
For the curve passes through (0, -2), we get c = 2,
Thus, the required particular solution is:-
Question 56.
A curve passes through the point (-1, 1) and at any point (x, y) of the curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4, -3). Find the equation of the curve.
Answer:
Given : 
⇒ 
⇒ 
⇒ 
The curve passes through (-2, 1)we have,
c = 0,
Question 57.
In a bank, principal increases at the rate fo r% per annum. Find the value of r if ` 100 double itself in 10 years.
(Given loge 2 = 0.6931)
Answer:
Given:
Here, p is the principal, r is the rate of interest per annum and t is the time in years.
Solving the differential equation we get,
⇒ 
⇒ 
⇒ 
As it is given that the principal doubles itself in 10 years, so
Let the initial interest be p1 (for t = 0 ), after 10 years p1 becomes 2p1.
Thus, 
for ( t = 0) …(i)
…(ii)
Substituting (i) in (ii), we get,
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Rate of interest = 6.931
Question 58.
In a bank, principal increases at the rate of 5% per annum. An amount of ` 1000 is deposited in the bank. How much will it worth after 10 years?
(Given e0.5 = 1.648)
Answer:
Given: rate of interest = 5%
P(initial) = Rs 1000
And,
⇒ 
⇒ 
⇒ 
For t = 0, we have p = 1000
For t = 10 years we have, 
Thus, principal is Rs1648 for t = 10 years.
Question 59.
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after t seconds.
Answer:
Given:
Volume V 
⇒ 
(constant)
⇒ 
⇒ 
⇒ 
For t = 0, r = 3 and for t = 3, r = 6, So, we have,
⇒ 
⇒ 
⇒ 
So after t seconds the radius of the balloon will be,
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Hence, radius of the balloon as a function of time is
Question 60.
In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
Answer:
Let y be the bacteria count, then, we have,
rate of growth of bacteria is proportional to the number present
So,
Where c is a constant,
Then, solving the equation we have,
Where k is constant of integration
And we have for t = 0, y = 10000,
…(i)
For t = 2hrs, y is increased by 10% i. e. y = 110000
⇒ 
from (i)
⇒ 
⇒ 
⇒ 
When y = 200000, we have,
⇒ 
⇒ 
⇒ 
⇒ 
Hence, t = 