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# Solve the Differential Equation - CBSE (Commerce) Class 12 - Mathematics

ConceptMethods of Solving First Order, First Degree Differential Equations Homogeneous Differential Equations

#### Question

Solve the differential equation :

y+x dy/dx=x−y dy/dx

#### Solution

y+x dy/dx=x−y dy/dx

x dy/dx + y dy/dx=x−y

⇒dy/dx=(x−y)/(x+y)    ......(1)

Let F(x, y) =(x−y)/(x+y)

F(λx, λy) = λF(x, y)
Therefore, F(x, y) is a homogeneous function of degree zero.

Let y=vx

dy/dx=v+x (dv)/dx

Substituting the value of y and dy/dx in (1) we get,

v + x (dv)/dx=(x−vx)/(x+vx)=(1−v)/(1+v)

x (dv)/dx=(1−v)/(1+v)−v=(1−v−v^2−v)/(1+v)=(1−2v−v^2)/(1+v)

(1+v)/(v^2+2v−1)dv=−dx/x

Integrating both sides, we have

1/2 log∣(y^2/x^2)+(2y)/x−1∣+log|x|=logc

⇒log∣(y^2/x^2)+(2y)/x−1∣+2log|x|=2logc

⇒log((y^2/x^2)+(2y)/x−1)(x^2)=logc^2

⇒((y^2+2yx−x^2)/x^2)(x^2) = c^2

⇒y^2+2yx−x^2=C           (where C=c^2)

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#### APPEARS IN

Solution Solve the Differential Equation Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations.
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