Question

(2y^2+3x)dx+4xydy=0
Using Exact method of differential equations

Answer 1

This ODE is exact because

$\dfrac{\partial(2y^2+3x)}{\partial y}=4y$

$\dfrac{\partial(4xy)}{\partial x}=4y$

So we're looking for a solution $\Psi(x,y)=C$ such that

$\dfrac{\partial\Psi}{\partial x}=2y^2+3x$

$\dfrac{\partial\Psi}{\partial y}=4xy$

Integrate both sides of the first PDE with respect to $x$:

$\displaystyle\int\frac{\partial\Psi}{\partial x}=\int(2y^2+3x)\,\mathrm dx$

$\Psi=2xy^2+3x+g(y)$

Differentiate both sides with respect to $y$:

$\dfrac{\partial\Psi}{\partial y}=4xy+\dfrac{\mathrm dg}{\mathrm dy}=4xy$

$\implies\dfrac{\mathrm dg}{\mathrm dy}=0\imples g(y)=C$

So the solution to the ODE is

$2xy^2+3x=C$