# ouchimdead's Question from Math Help Forum

#### Sudharaka

##### Well-known member
MHB Math Helper
Title: how do i show something is a solution without solving it? 2nd order DE

the wave equation

δ2y/δx2 = (1/c2)(δ2y/δt2)

show y(x,t) = f(x-ct)+g(x+ct) is a solution explicitly

please show me how, don't just say "substitution"
The only thing that you have to do here is to find the second derivatives of $$y(x,t)$$ with respect to $$x$$ and $$t$$. Then it could be easily seen that they satisfy the given differential equation.
$y(x,t) = f(x-ct)+g(x+ct)$
$\Rightarrow\frac{\partial}{\partial x}y(x,t) = \frac{\partial}{\partial x}f(x-ct)+\frac{\partial}{\partial x}g(x+ct)\mbox{ and }\frac{\partial}{\partial t}y(x,t) = -c\frac{\partial}{\partial t}f(x-ct)+c\frac{\partial}{\partial t}g(x+ct)$
$\Rightarrow\frac{\partial^{2}}{\partial x^2}y(x,t) = \frac{\partial^{2}}{\partial x^2}f(x-ct)+\frac{\partial^2}{\partial x^2}g(x+ct)\mbox{ and }\frac{\partial^2}{\partial t^2}y(x,t) = c^2\frac{\partial^2}{\partial t^2}f(x-ct)+c^2\frac{\partial^2}{\partial t^2}g(x+ct)$
$\therefore \frac{\partial^{2}}{\partial x^2}y(x,t)=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}y(x,t)$