Answer to Question #86617 in Differential Equations for Evol

Question

obtain the value of the constant c for which the function u(x,t) = cosαcxsinαt is a solution of the wave equation ∂ ^2u/∂t^2 = c^2∂ ^2u/∂ x^2

Answer 1

Solution:

If

$$u(x,t)=cos acx sin at(1)$$

is a solution of the wave equation

$$\frac{ ∂ ^2u}{∂t^2}=\frac{c^2∂ ^2u}{∂ x^2},(2)$$

then

$$\frac{ ∂ u}{∂t}=acos acx cos at,(3)$$

$$\frac{ ∂ ^2u}{∂t^2}=-a^2cos acx sin at,(4)$$

$$\frac{ ∂ u}{∂x}=-acsin acx sin at,(5)$$

$$\frac{ ∂ ^2u}{∂x^2}=-a^2c^2cos acx sin at,(6)$$

and

$$-a^2cos acx sin at=-a^2c^4cos acx sin at.(7)$$

Then from equation (7) we find c:

$$c^4=\frac{-a^2cos acx sin at}{-a^2cos acx sin at},$$

$$c^4=1,$$

$$c=1\\ {and }\\ c=-1.$$

Answer: c=1 and c=-1.