# How do you find all solutions of the differential equation (d^2y)/(dx^2)=x^-2?

Dec 29, 2016

$y = - \ln \left(\left\mid x \right\mid\right) + {C}_{1} x + {C}_{2}$

#### Explanation:

Note that when we integrate things like ${\mathrm{dx}}^{2}$ we will be left with $\mathrm{dx}$, so we will need to integrate this twice.

Separating variables:

${d}^{2} y = {x}^{-} 2 {\mathrm{dx}}^{2}$

Integrating:

$\int {d}^{2} y = \int {x}^{-} 2 {\mathrm{dx}}^{2}$

This is analogous to doing $\int \mathrm{dy} = \int {x}^{-} 2 \mathrm{dx} \implies y = - {x}^{-} 1 + {C}_{1}$, but a $\mathrm{dx}$ term will still be left.

$\mathrm{dy} = \left(- {x}^{-} 1 + {C}_{1}\right) \mathrm{dx}$

Integrating again:

$\int \mathrm{dy} = \int \left(- \frac{1}{x} + {C}_{1}\right) \mathrm{dx}$

$y = - \ln \left(\left\mid x \right\mid\right) + {C}_{1} x + {C}_{2}$