Answer to Question #192862 in Differential Equations for Sourav

Question

Determine all the first and second order partial derivatives for the function:

u(x,t)=Ce^(1-n²π²)t sin (nπx)

Answer 1

Let us determine all the first order partial derivatives for the function

$u(x,t)=Ce^{(1-n²π²)t} \sin (nπx):$

$\frac{{\partial}u(x,t)}{\partial x}=Cn\pi e^{(1-n²π²)t} \cos (nπx), \ \ \frac{{\partial}u(x,t)}{\partial t}=C(1-n²π²)e^{(1-n²π²)t} \sin (nπx).$

Let us determine all the second order partial derivatives for the function

$u(x,t)=Ce^{(1-n²π²)t} \sin (nπx):$

$\frac{{\partial}u^2(x,t)}{\partial x^2}=-Cn^2\pi^2 e^{(1-n²π²)t} \sin (nπx), \ \ \frac{{\partial}u^2(x,t)}{\partial t^2}=C(1-n²π²)^2e^{(1-n²π²)t} \sin (nπx),$

$\frac{{\partial}u^2(x,t)}{\partial x \partial t}=Cn\pi(1-n²π²) e^{(1-n²π²)t} \cos (nπx), \ \ \frac{{\partial}u^2(x,t)}{\partial t \partial x}=Cn\pi (1-n²π²)e^{(1-n²π²)t} \cos (nπx).$