Question

Find a solution to dy/dx=xy+9x+4y+36

If necessary, use K to denote an arbitrary constant.

Answer 1

The given equation can be written as, $\dfrac{dy}{dx} = (x+4)(y+9)$

Separating the variables, $\dfrac{dy}{y+9} = (x+4)dx$

Integrating we get,

$\begin{aligned} \log (y+9) &= \dfrac{x^{2}}{2}+4x + c\\ y+9 &= e^{\frac{1}{2}(x^2+8x+2c)}\\ y&= Ke^{\frac{1}{2}(x^2+8x)} -9,~~~\text{where } K = e^{c} \end{aligned}$