$Differential equations$

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$Differential equations$

Solve the ordinary differential equation $\frac{d^{2}y }{dx^{2} } =y$ for y as a function of x. (Note that this is equivalent to y" = y.)

Solve the partial differential equation $ \frac{∂^{2} f}{∂x∂y} = xy$ for f as a function of x and y.

Ordinary Differential Equations Partial Differential Equations

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1. The associated equation is $r^2-1=0$, with roots $r=\pm1$.
The general solution then is given by $y= c_1e^x+c_2e^{-x}$

2. Integrating with respect to $x$, $\frac{\partial f}{\partial y}=\frac{x^2y}{2}+h(y)$, for some function $h(y)$.

Integrating now with respect to $y$ gives the general solution: $f(x,y) = \frac{x^2y^2}{4} + H(y)+G(x)$,
where $H(y)$ and $G(x)$ are some differentiable functions.

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$Differential equations$

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