Q. 275.0( 1 Vote )

# Solve the differential equation: x2 dy + y(x + y)dx = 0, given that y = 1 when x = 1.

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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