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NCERT Solutions for Class 12 Maths Differential Equations
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve each of them.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 1.
(x2 + xy) dy = (x2 + y2) dx
NCERT Solutions:
Given differential equation is
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 2.
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 3.
(x – y) dy – (x + y) dx = 0
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 4.
(x2 – y2) dx + 2xy dy = 0
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 5.
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 6.
NCERT Solutions:
Given differential equation is
or 
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 7.
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 8.
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 9.
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to x, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 10.
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dx/dy is in form of g(x/y), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put x = vy …(2)
Differentiating the equation (2) with respect to y, we get
On substituting x = vy and in equation (1), we get
Integrating, we get
where C is an arbitrary constant. This is required general solution of the given differential equation.
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 11.
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
NCERT Solutions:
Given differential equation is (x + y) dy + (x – y) dx = 0.
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to y, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
…(3)
where C is an arbitrary constant. This is required general solution of the given differential equation.
It is given that y = 1 when x = 1.
From the equation (3), we get
Substituting the value of C in equation (3), we get
This is required particular solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 12.
x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
NCERT Solutions:
Given differential equation is x2 dy + (xy + y2) dx = 0.
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to y, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
…(3)
where C is an arbitrary constant. This is general solution of the given differential equation.
It is given that y = 1 when x = 1.
From the equation (3), we get
Substituting the value of C in equation (3), we get
This is required particular solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 13.
y = π/4 when x = 1
NCERT Solutions:
Given differential equation is
…(1)
In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to y, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
…(3)
where C is an arbitrary constant. This is general solution of the given differential equation.
It is given that y = π/4 when x = 1.
From the equation (3), we get
Substituting the value of C in equation (3), we get
This is required particular solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 14.
y = 0 when x = 1
NCERT Solutions:
Given differential equation is
…(1)
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to y, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
…(3)
where C is an arbitrary constant. This is general solution of the given differential equation.
It is given that y = 0 when x = 1.
From the equation (3), we get
Substituting the value of C in equation (3), we get
This is required particular solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 15.
y = 2 when x = 1
NCERT Solutions:
Given differential equation is
…(1)
To solve this, we need to put y = vx …(2)
Differentiating the equation (2) with respect to y, we get
On substituting y = vx and in equation (1), we get
Integrating, we get
…(3)
where C is an arbitrary constant. This is general solution of the given differential equation.
It is given that y = 2 when x = 1.
From the equation (3), we get
Substituting the value of C in equation (3), we get
This is required particular solution of the given differential equation.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 16.
A homogeneous differential equation of the from 
can be solved by making the substitution.
(A) y = vx
(B) v = yx
(C) x = vy
(D) x = v
NCERT Solutions:
To solve the homogeneous differential equation of the from , we need to put x = vy.
NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.5: Ques No 17.
Which of the following is a homogeneous differential equation?
(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
(B) (xy) dx – (x3 + y3) dy = 0
(C) (x3 + 2y2) dx + 2xy dy = 0
(D) y2 dx + (x2 – xy – y2) dy = 0
NCERT Solutions:
