Example. Find the general solution of the elliptic equation

$$
u_{x x}+4 u_{x y}+5 u_{y y}=0
$$

11 V.I. Afanasyev et al.
## Solution.

1. We transform equation (3) into its canonical form. We obtain

$$
u_{\xi \xi}+u_{\eta \eta}=0
$$

where $\xi=y-2 x, \eta=x$.

2. Equation (4) means that $u(\xi, \eta)$ is the real (or imaginary) part of an analytic function $f(\xi+\eta i)$. Therefore,

$$
u(x, y)=\operatorname{Re} f(y-2 x+x i)
$$

Answer. The general solution of equation (3) is

$$
u(x, y)=\operatorname{Re} f(y-2 x+x i)
$$

where $f(z)$ is an arbitrary analytic function.

### Conditions of the Problems. Find the general solution of the elliptic equation.

1. $u_{x x}+2 u_{x y}+5 u_{y y}=0$.
2. $u_{x x}-2 u_{x y}+3 u_{y y}=0$.
3. $u_{x x}+4 u_{x y}+5 u_{y y}=0$.
4. $u_{x x}-4 u_{x y}+8 u_{y y}=0$.
5. $u_{x x}+6 u_{x y}+13 u_{y y}=0$.
6. $u_{x x}-6 u_{x y}+18 u_{y y}=0$.
7. $u_{x x}+4 u_{x y}+20 u_{y y}=0$.
8. $u_{x x}-4 u_{x y}+40 u_{y y}=0$.
9. $u_{x x}-2 u_{x y}+26 u_{y y}=0$.
10. $u_{x x}+2 u_{x y}+10 u_{y y}=0$.

Answers.

1. $u=\operatorname{Re} f(y-x+2 x i)$.
2. $u=\operatorname{Re} f(y+x+\sqrt{2} x i)$.
3. $u=\operatorname{Re} f(y-2 x+x i)$.
4. $u=\operatorname{Re} f(y+2 x+2 x i)$.
5. $u=\operatorname{Re} f(y-3 x+2 x i)$.
6. $u=\operatorname{Re} f(y+3 x+3 x i)$.
7. $u=\operatorname{Re} f(y-2 x+4 x i)$.
8. $u=\operatorname{Re} f(y+2 x+6 x i)$.
9. $u=\operatorname{Re} f(y+x+5 x i)$.
10. $u=\operatorname{Re} f(y-x+3 x i)$.

## 5.5. Laplace's Equation in a Circle

### Problem Statement. Solve the Dirichlet boundary value problem for Laplace's equation in a circle:

$$
\begin{gathered}
\Delta u=0, \quad 0 \leqslant r<r_{0} \\
\left.u\right|_{r=r_{0}}=a \cos ^{3} \varphi+b \sin ^{3} \varphi+p \cos \varphi+q \sin \varphi+c
\end{gathered}
$$

### Plan of Solution. Laplace's equation (1) in polar coordinates $(r, \varphi)$ is

$$
\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \varphi^{2}}=0
$$

1. Find auxiliary solutions $v$ of equation (3) in the form

$$
v(r, \varphi)=R(r) \Phi(\varphi)
$$

where $|R(0)|<\infty$ and $\Phi(\varphi)$ is periodic with period $2 \pi$.

For this, we substitute the function $v(r, \varphi)$ into equation (3) and separate the variables. We obtain

$$
\frac{r \frac{d}{d r}\left(r \frac{d R}{d r}\right)}{R}=-\frac{\frac{d^{2} \Phi}{d \varphi^{2}}}{\Phi}=\lambda=\text { const }
$$

Therefore, the functions $R(r)$ and $\Phi(\varphi)$ are solutions of the related problems:

a) $\Phi^{\prime \prime}+\lambda \Phi=0, \Phi(\varphi+2 \pi)=\Phi(\varphi)$;

b) $r^{2} R^{\prime \prime}+r R^{\prime}-\lambda R=0,|R(0)|<\infty$.

2. Solve problem (a).

The general solution of the equation $\Phi^{\prime \prime}+\lambda \Phi=0$ is

$$
\Phi(\varphi)=\widetilde{A} e^{\sqrt{-\lambda} \varphi}+\widetilde{B} e^{-\sqrt{-\lambda} \varphi}
$$

It is periodic for $\lambda \geqslant 0$ and has a period of $2 \pi$ for $\lambda=n^{2}$ ( $n=0,1, \ldots$). We obtain:

$$
\Phi_{0}(\varphi)=\widetilde{A}_{0} \quad