How does one solve a fourth-order PDE of the form $\frac{\partial^4y}{\partial x^4}=c^2\frac{\partial^4y}{\partial t^4}$? It looks like a one dimensional wave equation, but I'm unfortunately very bad at PDEs. Would separating variables and using Fourier series work if I try hard enough?
[Math] Solving 4th-order PDE
partial differential equations
Best Answer
Since all terms of the PDE are in same order and constant coefficient, you can apply the similar technique that solving the wave equation:
$\dfrac{\partial^4y}{\partial x^4}=c^2\dfrac{\partial^4y}{\partial t^4}$
$\dfrac{\partial^4y}{\partial x^4}-c^2\dfrac{\partial^4y}{\partial t^4}=0$
$\left(\dfrac{\partial^4}{\partial x^4}-c^2\dfrac{\partial^4}{\partial t^4}\right)y=0$
$\left(\dfrac{\partial}{\partial x}-\sqrt{c}\dfrac{\partial}{\partial t}\right)\left(\dfrac{\partial}{\partial x}+\sqrt{c}\dfrac{\partial}{\partial t}\right)\left(\dfrac{\partial}{\partial x}-i\sqrt{c}\dfrac{\partial}{\partial t}\right)\left(\dfrac{\partial}{\partial x}+i\sqrt{c}\dfrac{\partial}{\partial t}\right)y=0$
$y(x,t)=f_1(t+\sqrt{c}x)+f_2(t-\sqrt{c}x)+f_3(t+i\sqrt{c}x)+f_4(t-i\sqrt{c}x)$