Question

10. (10 points) Solve the following ODE. (3y^2 + sec^2 (x)) dx +(6xy + y) dy = 0

Answer 1

The ODE is exact, since

`(3y^2+\sec^2x)_y=6y`

`(6xy+y)_x=6y`

so there is a solution of the form
`\Psi(x,y)=C` satisfying

`\Psi_x=3y^2+\sec^2x`

`\Psi_y=6xy+y`

Integrating both sides of the first PDE wrt
`x` gives

`\Psi=3xy^2+\tan x+f(y)`

and differentiating wrt
`y` gives

`\Psi_y=6xy+y=6xy+f'(y)\implies f'(y)=y\implies f(y)=\frac{y^2}2+C`

Then

`\Psi(x,y)=\boxed{3xy^2+\tan x+\frac{y^2}2=C}`