Solve (∂^2 u)/(∂ x^2) = (1)/(100)(∂^2 u)/(∂ t^2) 0

Solve $\frac{\partial^2 u}{\partial x^2} = \frac{1}{100} \frac{\partial^2 u}{\partial t^2}$ $0 < x < \pi$, $t > 0$ subject to the boundary conditions $u(0; t) = u(\pi; t) = 0$, $t > 0$ and initial condition $u(x; 0) = \frac{1}{10}\sin x$, and $\frac{\partial u}{\partial t}(x, 0) = 0$ $0 \le x \le \pi$. Answer: $u(x, t) = \frac{1}{10}\sin x \cos 10t$.

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00:01 Okay, we want to solve this differential equation subject to these boundary conditions and this initial condition.

00:10 So the form of the equation and the form of the initial conditions and the boundary conditions make me want to write this.

00:25 I'm just going to write a test form for the solution.

00:32 So a, b, c, d, and f are constants.

00:38 Let's take the time derivative.

00:49 So i know for the heat equation that it certainly makes sense to talk about products, sums of products of functions of x and functions of t.

01:00 And we know that, for instance, sine 2x goes to with e to the minus 4t as a solution of the left -hand side.

01:09 And sine of x times e to the minus t would also be a solution of the left -hand side.

01:14 Basically, what we might call a complementary solution.

01:17 The rest of these terms are to try and make a particular solution that will satisfy the equation.

01:33 And i'll take the time derivative and i'll take the second spatial derivative like that.

01:46 I'll substitute them in the equation.

02:23 Okay, and one thing i can see right away is that the f and d terms just cancel out.

02:32 And so what i'm left over with are the a, b, and c terms times sine x.

02:44 So i'm kind of doing this by undetermined coefficients, basically...

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