For each of the differential equations in question, find the particular solution satisfying the given condition:
x²dy + (xy + y²)dx = 0; y = 1 when x = 1

Question

For each of the differential equations in question, find the particular solution satisfying the given condition: x 2 dy + (xy + y 2 )dx = 0; y = 1 when x = 1

Philoid · Accepted Answer

x²dy + (xy + y²)dx = 0

Here, putting x = kx and y = ky

= k⁰.f(x,y)

Therefore, the given differential equation is homogeneous.

x²dy + (xy + y²)dx = 0

To solve it we make the substitution.

y = vx

Differentiating eq. with respect to x, we get

Integrating both sides, we get

y = 1 when x = 1

3x²y = y + 2x

y + 2x = 3x²y

The required solution of the differential equation.

For each of the differential equations in question, find the particular solution satisfying the given condition:x2dy + (xy + y2)dx = 0; y = 1 when x = 1

For each of the differential equations in question, find the particular solution satisfying the given condition:
x²dy + (xy + y²)dx = 0; y = 1 when x = 1