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Homogeneous Differential Equations

Class 12th Mathematics RS Aggarwal Solution
Exercise 20
  1. xdy = (x + y)dx In each of the following differential equation show that it is…
  2. (x2 - y2)dx + 2xydy = 0 In each of the following differential equation show…
  3. x2dy + y(x + y)dx = 0 In each of the following differential equation show that…
  4. (x - y)dy - (x + y)dx = 0 In each of the following differential equation show…
  5. (x + y)dy + (y - 2x)dx = 0 In each of the following differential equation show…
  6. (x2 + 3xy + y2)dx - x2dy = 0 In each of the following differential equation…
  7. 2xydx + (x2 + 2y2)dy = 0 In each of the following differential equation show…
  8. {dy}/{dx} + frac {x-2y}/{2x-y} = 0 In each of the following differential…
  9. {dy}/{dx} + frac { x^{2} - y^{2} }/{3xy} = 0 In each of the following…
  10. {dy}/{dx} = frac { x^{2} + y^{2} }/{2xy} In each of the following differential…
  11. {dy}/{dx} = frac {2xy}/{ ( x^{2} - y^{2} ) } In each of the following…
  12. x^{2} {dy}/{dx} = 2xy+y^{2} In each of the following differential equation…
  13. x2 {dy}/{dx} = x2 + xy + y2 In each of the following differential…
  14. y^{2} + ( x^{2} - xy ) {dy}/{dx} = 0 In each of the following differential…
  15. x {dy}/{dx} - y = 2 root { y^{2} - x^{2} } In each of the following…
  16. y^{2} dx + ( x^{2} + xy+y^{2} ) dy = 0 In each of the following differential…
  17. (x - y) {dy}/{dx} = x + 3y In each of the following differential…
  18. (x3 + 3xy2)dx + (y3 + 3x2y)dy = 0 In each of the following differential…
  19. ( x - root {xy} ) dy = ydx In each of the following differential equation…
  20. x2 {dy}/{dx} + y2 = xy In each of the following differential equation…
  21. x {dy}/{dx} = y (log y - log x + 1) In each of the following differential…
  22. x {dy}/{dx} –y + xsin {y}/{x} = 0 In each of the following…
  23. x {dy}/{dx} = y - xcos2 ( {y}/{x} ) In each of the following…
  24. ( xcos {y}/{x} ) frac {dy}/{dx} = ( ycos frac {y}/{x} ) + x In each of the…
  25. Find the particular solution of the different equation.2xy + y2 - 2x2…
  26. Find the particular solution of the differential equation { xsin^{2}…
  27. Find the particular solution of the differential equation {dy}/{dx} = frac…
  28. Find the particular solution of the differential equation xey/x - y + x…
  29. Find the particular solution of the differential equation xey/x - y + x…
  30. The slope of the tangent to a curve at any point (x,y) on it is given by…
Exercise 20
Question 1.

In each of the following differential equation show that it is homogeneous and solve it.

xdy = (x + y)dx
Answer:

Xdy = (x + y)dx





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:




Resubstituting the value of y = vx we get




Ans:
Question 2.

In each of the following differential equation show that it is homogeneous and solve it.

(x2 - y2)dx + 2xydy = 0
Answer:

(x2 - y2)dx + 2xydy = 0




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx









Integrating both the sides we get:




Resubstituting the value of y = vx we get





Ans: x2 + y2 = cx
Question 3.

In each of the following differential equation show that it is homogeneous and solve it.

x2dy + y(x + y)dx = 0
Answer:

x2dy + y(x + y)dx = 0




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:







Resubstituting the value of y = vx we get




⇒ x2y = c2(y + 2x)


Ans: x2y = c2(y + 2x)
Question 4.

In each of the following differential equation show that it is homogeneous and solve it.

(x - y)dy - (x + y)dx = 0
Answer:

(x - y)dy - (x + y)dx = 0





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:





Resubstituting the value of y = vx we get




Ans:



Question 5.

In each of the following differential equation show that it is homogeneous and solve it.

(x + y)dy + (y - 2x)dx = 0
Answer:

(x + y)dy + (y - 2x)dx = 0




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:




Resubstituting the value of y = vx we get




⇒ y2 + 2xy - 2x2 = c


Ans: y2 + 2xy - 2x2 = c
Question 6.

In each of the following differential equation show that it is homogeneous and solve it.

(x2 + 3xy + y2)dx - x2dy = 0
Answer:

(x2 + 3xy + y2)dx - x2dy = 0




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:







Resubstituting the value of y = vx we get
Question 7.

In each of the following differential equation show that it is homogeneous and solve it.

2xydx + (x2 + 2y2)dy = 0
Answer:

2xydx + (x2 + 2y2)dy = 0




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ 3x2y + 2y3 = C


Ans: 3x2y + 2y3 = C
Question 8.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx









Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ (y - x) = C(y + x)3


Ans: (y - x) = C(y + x)3
Question 9.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ (x2 + 2y2)3 = Cx2


Ans: (x2 + 2y2)3 = Cx2
Question 10.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ (x2 - y2) = cx


Ans: (x2 - y2) = cx
Question 11.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ y = C(y2 + x2)


Ans: y = C(y2 + x2)
Question 12.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:








Resubstituting the value of y = vx we get




Ans:



Question 13.

In each of the following differential equation show that it is homogeneous and solve it.

x2 = x2 + xy + y2
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:



⇒ tan - v = ln|x| + c


Resubstituting the value of y = vx we get


⇒ tan - (y/x) = ln|x| + c


Ans: tan - (y/x) = ln|x| + c
Question 14.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put x = vy







Integrating both the sides we get:





⇒ y = x(ln|y| + c)


Ans: y = x(ln|y| + c)
Question 15.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ y + = C|x|3


Ans: y + = C|x|3
Question 16.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put x = vy







Integrating both the sides we get:



Resubstituting the value of x = vy we get


⇒ log = C


Ans: log = C
Question 17.

In each of the following differential equation show that it is homogeneous and solve it.

(x - y) = x + 3y
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:





Resubstituting the value of y = vx we get



⇒ log|x + y| + = C


Ans: log|x + y| + = C
Question 18.

In each of the following differential equation show that it is homogeneous and solve it.

(x3 + 3xy2)dx + (y3 + 3x2y)dy = 0
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ y4 + 6x2y2 + x4 = C


Ans: y4 + 6x2y2 + x4 = C
Question 19.

In each of the following differential equation show that it is homogeneous and solve it.

dy = ydx
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx







Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ 2 = C


Ans: 2 = C
Question 20.

In each of the following differential equation show that it is homogeneous and solve it.

x2 + y2 = xy
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get






Ans:



Question 21.

In each of the following differential equation show that it is homogeneous and solve it.

(log y - log x + 1)
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:






Resubstituting the value of y = vx we get



Ans:
Question 22.

In each of the following differential equation show that it is homogeneous and solve it.

x–y + xsin = 0
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



⇒ Xtan = C


Ans: Xtan = C
Question 23.

In each of the following differential equation show that it is homogeneous and solve it.

x = y - xcos2
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



Ans:



Question 24.

In each of the following differential equation show that it is homogeneous and solve it.
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:



⇒ sinv = ln|x| + c


Resubstituting the value of y = vx we get



Ans:
Question 25.

Find the particular solution of the different equation.2xy + y2 - 2x2 = 0, it being given that y = 2 when x = 1
Answer:

2xy + y2 - 2x2 = 0




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



Now,


y = 2 when x = 1





Ans:



Question 26.

Find the particular solution of the differential equation dx + xdy = 0, it being given that y = when x = 1.
Answer:



⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



y = when x = 1



⇒ c = 1


Ans:



Question 27.

Find the particular solution of the differential equation given that y = 1 when x = 1.
Answer:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx








⇒


Integrating both the sides we get:




Resubstituting the value of y = vx we get



y = 1 when x = 1


1 + 0 = - 0 + c


⇒ c = 1


⇒ = 1


Ans: = 1
Question 28.

Find the particular solution of the differential equation xey/x - y + x = 0, given that y(1) = 0.
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



Now,y(1) = 0




= 1


Ans: = 1
Question 29.

Find the particular solution of the differential equation xey/x - y + x = 0, given that y(e) = 0.
Answer:





⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



Now,y(e) = 0




y = - xlog(log|x|)


Ans: y = - xlog(log|x|)
Question 30.

The slope of the tangent to a curve at any point (x,y) on it is given by , where x>0 and y>0. If the curve passes through the point, find the equation of the curve.
Answer:

It is given that:




⇒ the given differential equation is a homogenous equation.


The solution of the given differential equation is :


Put y = vx






Integrating both the sides we get:




Resubstituting the value of y = vx we get



the curve passes through the point




⇒ sec


Ans:The equation of the curve is: sec