Example 6. Integrate the equation $y y^{\prime \prime}-y^{\prime 2}=0$.
Solution. This is an equation of the form (10.12), as it does not explicitly contain the argument $x$. Let $y^{\prime}=p$, then

$$
y^{\prime \prime}=\frac{d y^{\prime}}{d x}=\frac{d y^{\prime}}{d y} \cdot \frac{d y}{d x}=\frac{d p}{d y} \cdot p, y^{\prime \prime}=\frac{d p}{d y} \cdot p
$$

Substituting the expressions for $y^{\prime}$ and $y^{\prime \prime}$ into the original equation, we obtain a first-order equation with separable variables $y$ and $p$:

$$
y p \frac{d p}{d y}-p^{2}=0
$$

Separating the variables, we get

$$
\frac{d p}{p}-\frac{d y}{y}=0(p y \neq 0)
$$

Integrating, we find

$$
\ln |p|=\ln |y|+\ln \left|C_{1}\right|
$$

from which $p=C_{1} y$.

Since $p=y^{\prime}$, then $\frac{d y}{d x}=C_{1} y$. Integrating this equation, we get $\ln |y|=C_{1} x+\ln \left|C_{2}\right|$ or $y=C_{2} e^{C_{1} x}$.

Remark. The solutions $y=0, y=C\left(y^{\prime}=0\right)$ are obtained from the general solution respectively when $C_{2}=0$ and $C_{1}=0$.

## Problems

Integrate the second-order differential equations:

1. $y^{\prime \prime}=\sin x$.
2. $y^{\prime \prime}=-\frac{1}{x^{2}}$.
3. $y^{\prime \prime}=\frac{6}{y^{3}}$.
4. $4 y^{\prime \prime} \sqrt{y}=1$.
5. $y^{\prime \prime}=\frac{1}{a}\left(\sqrt{1+y^{\prime 2}}\right)^{3}$.
6. $x y^{\prime \prime}-y^{\prime}=0$.
7. $\left(1+x^{2}\right) y^{\prime \prime}+y^{\prime 2}+1=0$.
8. $y y^{\prime \prime}+y^{\prime 2}+1=0$.

## Answers

1. $y=-\sin x+C_{1} x+C_{2}$.
2. $y=\ln x+C_{1} x+C_{2}$.
3. $C_{1} y^{2}-6=$
$=\left(C_{1} x+C_{2}\right)^{2}$.
4. $3 x+C_{1}=4 \sqrt{C+\sqrt{y}}(\sqrt{y}-2 C)$.
5. $\left(x-C_{1}\right)^{2}+$ $+\left(y-C_{2}\right)^{2}=a^{2}$.
6. $y=C_{1} x^{2}+C_{2}$.
7. $y=-C_{1} x+\left(C_{1}^{2}+1\right) \ln \left(C_{1}+x\right)+$ $+C_{2} \cdot 8 .\left(x+C_{2}\right)^{2}+y^{2}=C_{1}^{2}$.

## § 10.2. Homogeneous Second-Order Differential Equations with Constant Coefficients

A second-order differential equation with constant coefficients is an equation of the form

$$
a y^{\prime \prime}+b y^{\prime}+c y=f(x),
$$

where $a, b, c$ are constants, and $f(x)$ is a function of $x$.

If $f(x) \equiv 0$, then equation (10.13) is called a homogeneous differential equation with constant coefficients or an equation without a right-hand side:

$$
a y^{\prime \prime}+b y^{\prime}+c y=0 \text {. }
$$

Equation (10.14) can be written as

$$
y^{\prime \prime}+p y^{\prime}+q y=0
$$

The equation

$$
k^{2}+p k+q=0
$$

is called the characteristic equation for equation (10.15).

Depending on the roots $k_{1}$ and $k_{2}$ of the characteristic equation (10.16), we obtain the general solution of equation (10.15) in the form

$$
y=C_{1} e^{k_{1} x}+C_{2} e^{k_{2} x}
$$

if $k_{1}$ and $k_{2}$ are distinct real numbers;

$$
y=\left(C_{1} x+C_{2}\right) e^{k_{1} x}
$$

if $k_{1}=k_{2}$

$$
y=e^{\alpha x}\left(C_{1} \cos \beta x+C_{2} \sin \beta x\right)
$$

if $k_{1}=\alpha+i \beta, k_{2}=\alpha-i \beta-$ are complex numbers.