Example. Solve the Dirichlet boundary value problem for Poisson's equation in a sphere

$$
\begin{gathered}
\Delta u=14 x y, \quad 0 \leqslant r<2 \\
\left.u\right|_{r=2}=14
\end{gathered}
$$
Solution.

1. The particular solution of equation (6) is sought in the form

$$
u_{\text {p }}=E\left(x^{2}+y^{2}+z^{2}\right) x y
$$

Substituting it into equation $(6)$, we get

$$
6 E x y+6 E x y+2 E x y=14 x y
$$

Therefore, $E=1$ and

$$
u_{\text {p }}=\left(x^{2}+y^{2}+z^{2}\right) x y
$$

## 2. The function

$$
v(x, y, z)=u(x, y, z)-\left(x^{2}+y^{2}+z^{2}\right) x y
$$

satisfies the Laplace equation

$$
\Delta u=0, \quad 0 \leqslant r<2
$$

and the boundary condition

$$
\left.v\right|_{r=2}=14-\left.4 x y\right|_{r=2}
$$

3. It is easy to verify that the function $w(x, y, z)=14-4 x y$ is a solution to the Dirichlet problem (9)-(10). By the uniqueness theorem for the solution of the Dirichlet problem for the Laplace equation in a sphere, there are no other solutions.
4. Substituting $v(x, y, z)$ into equation (8), we get

$$
u(x, y, z)=14-4 x y+\left(x^{2}+y^{2}+z^{2}\right) x y
$$

Answer. $u(x, y, z)=\left(x^{2}+y^{2}+z^{2}-4\right) x y+14$.

Conditions of the problems. Solve the boundary value problem of Dirichlet for the Poisson equation in a sphere.

1. $u_{x x}+u_{y y}+u_{z z}=14 x y+7 x z, 0 \leqslant r<1,\left.u\right|_{r=1}=5$.
2. $u_{x x}+u_{y y}+u_{z z}=7 x y-7 x z+14 y z, 0 \leqslant r<2,\left.\quad u\right|_{r=2}=1$.
3. $u_{x x}+u_{y y}+u_{z z}=-7 x y+14 x z, 0 \leqslant r<2,\left.u\right|_{r=2}=4$.
4. $u_{x x}+u_{y y}+u_{z z}=14 x y-14 y z, 0 \leqslant r<3, u_{r=3}=8$.
5. $u_{x x}+u_{y y}+u_{z z}=28 x y-14 x z+14 y z, 0 \leqslant r<2,\left.u\right|_{r=2}=7$.
6. $u_{x x}+u_{y y}+u_{z z}=28 y z-7 x z, 0 \leqslant r<3,\left.u\right|_{r=3}=6$.
7. $u_{x x}+u_{y y}+u_{z z}=28 x y-14 x z+14 y z, 0 \leqslant r<4,\left.u\right|_{r=4}=7$.
8. $u_{x x}+u_{y y}+u_{z z}=14 x y-7 x y+7 y z, 0 \leqslant r<1,\left.u\right|_{r=1}=1$.
9. $u_{x x}+u_{y y}+u_{z z}=28 x y-14 x z-7 y z, 0 \leqslant r<2,\left.u\right|_{r=2}=4$.
10. $u_{x x}+u_{y y}+u_{z z}=-28 x y+28 x z-14 y z, 0 \leqslant r<3,\left.u\right|_{r=3}=5$.

Answers.

1. $u(x, y, z)=5-x y-x z / 2+\left(x^{2}+y^{2}+z^{2}\right)(x y+x z / 2)$.
2. $u(x, y, z)=1-2 x y+2 x z+\left(x^{2}+y^{2}+z^{2}\right)(x y / 2-x z / 2+y z)$.
3. $u(x, y, z)=4+2 x z-4 x z+\left(x^{2}+y^{2}+z^{2}\right)(x y / 2+x z)$.
4. $u(x, y, z)=8-9 x y+9 y z+\left(x^{2}+y^{2}+z^{2}\right)(x y-y z)$.
5. $u(x, y, z)=7-8 x y+4 x z-4 y z+\left(x^{2}+y^{2}+z^{2}\right)(2 x y-x z+y z)$.
6. $u(x, y, z)=6-9 x y / 2-18 y z+\left(x^{2}+y^{2}+z^{2}\right)(-x y / 2+2 y z)$.
7. $u(x, y, z)=7-8 x y+x z-4 y z+\left(x^{2}+y^{2}+z^{2}\right)(2 x y-x z+y z)$.
8. $u(x, y, z)=1-x y+x z / 2-y z / 2+\left(x^{2}+y^{2}+z^{2}\right)(2 x y-x z / 2+y z / 2)$.
9. $u(x, y, z)=4-8 x y+4 x z+2 y z+\left(x^{2}+y^{2}+z^{2}\right)(2 x y-x z-y z / 2)$.
10. $u(x, y, z)=5+18 x y-18 x z+9 y z+\left(x^{2}+y^{2}+z^{2}\right)(-2 x y+2 x z-y z)$.

### 5.14. Homogeneous Wave Equation on an Interval

Problem Statement. Solve the first mixed problem for the homogeneous wave equation on an interval

$$
\begin{aligned}
& u_{t t}-a^{2} u_{x x}=0, \quad x \in(0, l), \quad t \in(0, \infty) \\
& u(x, 0)=f(x), \quad u_{t}(x, 0)=g(x), \quad x \in(0, l) \\
& u(0, t)=u(l, t)=0, \quad t \in(0, \infty)
\end{aligned}
$$

SOLUTION PLAN.

1. Find auxiliary solutions $v(x, t)$ of equation (1) in the form

$$
v(x, t)=X(x) T(t)
$$

where $v(0, t)=v(l, t)=0$, i.e., $X(0)=X(l)=0$. For this, we substitute $v(x, t)=X(x) T(t)$ into equation (1) and separate the variables. We get

$$
\frac{X^{\prime \prime}}{X}=\frac{T^{\prime \prime}}{a^{2} T}=\lambda=\text { const. }
$$

Therefore, the functions $X(x)$ and $T(t)$ are solutions of the related problems:

a) $X^{\prime \prime}(x)-\lambda X(x)=0, \quad X(0)=X(l)=0$

b) $T^{\prime \prime}-\lambda a^{2} T=0$.

3. Solve problem (a).

The equation $X^{\prime \prime}-\lambda X=0$ has the general solution

$$
X(x)=