# Linear 2nd order diff. eqn.

1. Oct 7, 2008

### musemonkey

1. This is exercise 3.19.15 from Boyce & DiPrima's Differential Equations.

$$u'' + u = F(t), u(0) = 0, u'(0) = 0$$

$$0 \leq t < \pi , F(t) = F_0t$$
$$\pi \leq t \leq 2\pi , F(t) = F_0(2\pi-t)$$
$$t > 2\pi , F(t) = 0$$

Solve for $$u(t)$$.

2. The solution to the homogenous equation is

$$u'' + u = 0$$

is $$u(t) = c_1 cos(t) + c_2 sin(t)$$.

3. Plugging in the boundary conditions yields

$$u(0) = c_1 + 0 = 0$$
$$u'(t) = - c_1 sin(t) + c_2 cos(t)$$
$$u'(0) = c_2 = 0$$.

Then a particular solution of the form $$u_p = A t$$ works

$$u'' + u = At = F_0t$$,

and so the solution for $$0 \leq t < \pi$$ I find to be $$u_1(t) = F_0 t$$. But the solution in the back of the textbook states $$u_1(t) = F_0t - F_0 sin(t)$$, which clearly also satisfies the differential equation and the initial value conditions, but it's inconsistent with what I get. Where did I go wrong?