Q. 275.0( 1 Vote )

Solve the differe

Answer :

Given, A differential equation x2 dy + y(x + y)dx = 0

To Find: Find the particular solution at y = 1 and x = 1


Explanation: We have given


x2 dy + y(x + y)dx = 0


x2 dy = - y(x + y)dx




…(i)


Let F(x, y) =


To check that, given differential equation is homogenous ,


Put x = λx and y = λy in F(x, y)


Then,



Now, Taking λ2 common from both numerator and denominator



In F(x, y) If λ0, then F(x, y) is a homogenous function of degree zero.


Let’s put y = vx in equation(i)


On differentiating y , we get


…(ii)


Now, Compare the equation (i) and (ii)



Taking x2 as common from the numerator and denominator






Now, Integrating both sides




Solve dv by completing the square method




We know, , then








Now, putting






Now, Put x = 1 and y = 1




1 = 3C



Put the value of C




Hence,


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