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,