Example 5. Solve the equation

$$
y^{\mathrm{IV}}+4 y^{\prime \prime \prime}+8 y^{\prime \prime}+8 y^{\prime}+4 y=0
$$
Solution. We form the characteristic equation

$$
\lambda^{4}+4 \lambda^{3}+8 \lambda^{2}+8 \lambda+4=0
$$

or

$$
\left(\lambda^{2}+2 \lambda+2\right)^{2}=0
$$

It has double complex roots $\lambda_{1}=\lambda_{2}=-1-i, \lambda_{3}=\lambda_{4}=-1+i$ and, consequently, the general solution will have the form

$$
y_{0.0}=C_{1} e^{-x} \cos x+C_{2} e^{-x} \sin x+C_{3} x e^{-x} \cos x+C_{4} x e^{-x} \sin x
$$

or

$$
y_{0.0}=e^{-x}\left(C_{1}+C_{3} x\right) \cos x+e^{-x}\left(C_{2}+C_{4} x\right) \sin x
$$

$\square$

## Problems for Independent Solution

Form linear homogeneous differential equations given their characteristic equations:

410. $9 \lambda^{2}-6 \lambda+1=0$.
411. $\lambda(\lambda+1)(\lambda+2)=0$.
412. $\lambda^{2}+3 \lambda+2=0$
413. $\left(\lambda^{2}+1\right)^{2}=0$.
414. $2 \lambda^{2}-3 \lambda-5=0$.
415. $\lambda^{3}=0$.

Form linear homogeneous differential equations if the roots of their characteristic equations are known, and write their general solutions:

416. $\lambda_{1}=1, \lambda_{2}=2$.
417. $\lambda_{1}=1, \lambda_{2}=1$.
418. $\lambda_{1}=3-2 i, \quad \lambda_{2}=3+2 i$.
419. $\lambda_{1}=1, \lambda_{2}=1, \quad \lambda_{3}=1$.

Form linear homogeneous differential equations if their fundamental systems of solutions are given:
420. $e^{-x}, e^{x}$.
421. $1, e^{x}$.
422. $e^{-2 x}, x e^{-2 x}$.
423. $\sin 3 x, \cos 3 x$.
424. $1, x$
425. $e^{x}, e^{2 x}, e^{3 x}$.
426. $e^{x}, x e^{x}, x^{2} e^{x}$.
427. $e^{x}, x e^{x}, e^{2 x}$.
428. $1, x, e^{x}$.
429. $1, \sin x, \cos x$.
430. $e^{2 x}, \sin x, \cos x$.
431. $1, e^{-x} \sin x, e^{-x} \cos x$.

Integrate the following equations and, where specified, solve the Cauchy problem.

432. $y_{\prime \prime \prime}^{\prime \prime}-y=0$.
433. $3 y^{\prime \prime}-2 y^{\prime}-8 y=0$.
434. $y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0$,
435. $y^{\prime \prime}+2 y^{\prime}+y=0$.

$y(0)=1, \quad y^{\prime}(0)=2, y^{\prime \prime}(0)=3$.

436. $y^{\prime \prime}-4 y+3 y=0$,
437. $y^{\prime \prime \prime}+6 y^{\prime \prime}+11 y^{\prime}+6 y=0$.

$$
y(0)=6, \quad y^{\prime}(0)=10
$$

438. $y^{\prime \prime}-2 y^{\prime}-2 y=0$.
439. $y^{\mathrm{v} 1}+2 y^{\mathrm{v}}+y^{\mathrm{Nv}}=0$.
440. $4 y^{\prime \prime}-8 y^{\prime}+5 y=0$.
441. $y^{\prime \prime \prime}-8 y=0$.
442. $y^{\text {IV }}+4 y^{\prime \prime \prime}+10 y^{\prime \prime}+12 y^{\prime}+5 y=0$.
443. $y^{\prime \prime}-2 y^{\prime}+2 y=0$,

$$
y(0)=0, \quad y^{\prime}(0)=1
$$

444. $y^{\prime \prime}-2 y^{\prime}+3 y=0$,
445. $y^{\text {IV }}+2 y^{\prime \prime \prime}+4 y^{\prime \prime}-2 y^{\prime}-5 y=0$.

$$
y(0)=1, \quad y^{\prime}(0)=3
$$

446. $y^{\vee}+4 y^{\mathrm{Iv}}+5 y^{\prime \prime \prime}-6 y^{\prime}-4 y=0$.
447. $y^{\prime \prime \prime}+2 y^{\prime \prime}-y^{\prime}-2 y=0$.
448. $y^{\prime \prime \prime}-2 y^{\prime \prime}+2 y^{\prime}=0$.
449. $y^{\mathrm{IV}}-y=0$.
450. $y^{x}=0$.
451. $y^{\prime \prime \prime}-3 y^{\prime}-2 y=0$.
452. $2 y^{\prime \prime \prime}-3 y^{\prime \prime}+y^{\prime}=0$.
453. $y^{\prime \prime \prime}+y^{\prime \prime}=0$,

$y(0)=1, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1$.

$3^{\circ}$. Linear non-homogeneous equations with constant coefficients. A. Method of undetermined coefficients. Let the differential equation be given by

$$
a_{0} y^{(n)}+a_{1} y^{(n-1)}+\ldots+a_{n} y=f(x)
$$

with constant real coefficients $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$.

Theorem. The general solution of the non-homogeneous equation (11) is equal to the sum of the general solution of the corresponding homogeneous equation and any particular solution of the non-homogeneous equation.

Finding the general solution of the corresponding homogeneous equation is carried out according to the rules outlined in section 2. Thus, the problem of integrating equation (11) is reduced to finding a particular solution $y_{4 . \text { n }}$ of the non-homogeneous equation. In the general case, the integration of equation (11) can be performed using the method of variation of arbitrary constants (see below section $5^{\circ}$ ). For right-hand sides of a special form, the solution is often found more easily using the so-called method of undetermined coefficients. The general form of the right-hand side $f(x)$ of equation (11), for which the method of undetermined coefficients can be applied, is as follows:

$$
f(x)=e^{\alpha x}\left[P_{l}(x) \cos \beta x+Q_{m}(x) \sin \beta x\right]
$$

where $P_{l}(x)$ and $Q_{m}(x)$ are polynomials of degree $l$ and $m$ respectively. In this case, the particular solution $y_{\text {ch.n }}$ of equation (11) is sought in the form

$$
y_{4, H}=x^{0} e^{\alpha x}\left[\widetilde{P}_{k}(x) \cos \beta x+\widetilde{Q}_{k}(x) \sin \beta x\right]
$$

where $k=\max (m, l), \widetilde{P}_{k}(x)$ and $\widetilde{Q}_{k}(x)$ are polynomials of degree $k$ in $x$ with undetermined coefficients, and $s$ is the multiplicity of the root $\lambda=\alpha+i \beta$ of the characteristic equation (if $\alpha \pm i \beta$ is not a root of the characteristic equation, then $s=0$ ).