Example 4. Knowing the fundamental system of solutions $y_{1}=\ln x, y_{2}=x$ of the corresponding homogeneous equation, find a particular solution of the equation

$$
x^{2}(1-\ln x) y^{\prime \prime}+x y^{\prime}-y=\frac{(1-\ln x)^{2}}{x}
$$

satisfying the condition $\lim _{x \rightarrow+\infty} y=0$.
Solution. Applying the method of variation of constants, we find the general solution of equation (39):

$$
y=C_{1} \ln x+C_{2} x+\frac{1-2 \ln x}{4 x}
$$

As $x \rightarrow+\infty$, the first two terms on the right-hand side of (40) tend to infinity, and for any $C_{1}, C_{2}$, not both zero, the function $C_{1} \ln x+C_{2} x$ is an infinitely large function as $x \rightarrow+\infty$. The third term on the right-hand side of (40) has a limit of zero as $x \rightarrow+\infty$, which can be easily established using L'Hôpital's rule. Thus, the function $y=\frac{1-2 \ln x}{4 x}$, which is obtained from (40) when $C_{1}=0$ and $C_{2}=0$, will be a solution of equation (39) satisfying the condition $\lim _{x \rightarrow+\infty} y=0$.

## Problems for Independent Solution

Integrate the following equations, given one particular solution $y_{1}$ of the homogeneous equation.

636. $(2 x+1) y^{\prime \prime}+(4 x-2) y^{\prime}-8 y=0 ; \quad y_{1}=e^{m x}$.
637. $\left(x^{2}-x\right) y^{\prime \prime}+(2 x-3) y^{\prime}-2 y=0 ; \quad y_{1}$ - a rational fraction, with linear factors in the denominator - divisors of the coefficient of $y^{\prime \prime}$.
638. $\left(3 x+2 x^{2}\right) y^{\prime \prime}-6(1+x) y^{\prime}+6 y=6 ; \quad y_{1}$ - a polynomial.
639. $x^{2}(\ln x-1) y^{\prime \prime}-x y^{\prime}+y=0 ; \quad y_{1}=x$.
640. $y^{\prime \prime}+(\operatorname{tg} x-2 \operatorname{ctg} x) y^{\prime}+2 \operatorname{ctg}^{2} x \cdot y=0 ; \quad y_{1}=\sin x$.
641. $y^{\prime \prime}+\operatorname{tg} x \cdot y^{\prime}+\cos ^{2} x \cdot y=0 ; \quad y_{1}=\cos (\sin x)$.
642. $\left(1+x^{2}\right) y^{\prime \prime}+x y^{\prime}-y+1=0 ; \quad y_{1}=x$.

6.43. $x^{2} y^{\prime \prime}-x y^{\prime}-3 y=5 x^{4} ; \quad y_{1}=\frac{1}{x}$.

644. $(x-1) y^{\prime \prime}-x y^{\prime}+y=(x-1)^{2} e^{x} ; \quad y_{1}=e^{x}$.
645. $y^{\prime \prime}+y^{\prime}+e^{-2 x} y=e^{-3 x} ; \quad y_{1}=\cos e^{-x}$.
646. $\left(x^{4}-x^{3}\right) y^{\prime \prime}+\left(2 x^{3}-2 x^{2}-x\right) y^{\prime}-y=\frac{(x-1)^{2}}{x} ; \quad y_{1}=\frac{1}{x}$.
647. $y^{\prime \prime}-y^{\prime}+y e^{2 x}=x e^{2 x}-1 ; \quad y_{1}=\sin e^{x}$.
648. $x(x-1) y^{\prime \prime}-(2 x-1) y^{\prime}+2 y=x^{2}(2 x-3) ; \quad y_{1}=x^{2}$.
649. A chain 6 meters long, with 1 meter hanging off a table, begins to slide down without friction. How long will it take for the entire chain to slide off?
650. Find the equation of motion of a point if the acceleration as a function of time is given by the formula $a=1.2 \cdot t$ and if at $t=0$ the distance $s=0$, and at $t=5$ the distance $s=20$.
651. A body of mass $m$ slides on a horizontal plane under the impulse that gave it an initial velocity $v_{0}$. A friction force of $-\mathrm{km}$ acts on the body. Find the distance the body can travel.
652. A material point of mass $m=1$ moves in a straight line, approaching a center that repels it with a force equal to $k^{2} x$ (where $x$ is the distance of the point from the center). At $t=0, x=a, \frac{d x}{d t}=k a$. Find the law of motion.

Integrate the following equations using the method of variation of constants:

653. $y^{\prime \prime}+y=\frac{1}{\sin x}$.
654. $y^{\prime \prime}-y^{\prime}=\frac{1}{e^{x}+1}$.
655. $y^{\prime \prime}+y=\frac{1}{\cos ^{3} x}$.
656. $y^{\prime \prime}+y=\operatorname{ctg} x$.
657. $y^{\prime \prime}-2 y^{\prime}+y=\frac{e^{x}}{x^{2}+1}$.
658. $y^{\prime \prime}+2 y^{\prime}+2 y=\frac{1}{e^{x} \sin x}$.
659. $y^{\prime \prime}+y=\frac{2}{\sin ^{3} x}$.
660. $y^{\prime \prime}-y^{\prime}=e^{2 z} \cos e^{2}$.
661. $y^{\prime \prime \prime}+y^{\prime \prime}=\frac{x-1}{x^{2}}$.
662. $x y^{\prime \prime}-\left(1+2 x^{2}\right) y^{\prime}=4 x^{3} e^{a^{2}}$.
663. $y^{\prime \prime}-2 y^{\prime} \operatorname{tg} x=1$.
664. $x \ln x \cdot y^{\prime \prime}-y^{\prime}=\ln ^{2} x$.
665. $x y^{\prime \prime}+(2 x-1) y^{\prime}=-4 x^{2}$.
666. $y^{\prime \prime}+y^{\prime} \operatorname{tg} x=\cos x \operatorname{ctg} x$.

Find the solutions of the following differential equations under the given conditions at infinity:

667. $4 x y^{\prime \prime}+2 y^{\prime}+y=1, \quad \lim _{x \rightarrow+\infty} y=1 ; \quad y_{1}=\sin \sqrt{x}, \quad y_{2}=\cos \sqrt{x}$.
668. $4 x y^{\prime \prime}+2 y^{\prime}+y=\frac{6+x}{x^{2}}, \quad \lim _{x \rightarrow+\infty} y=0$.
669. $\left(1+x^{2}\right) y^{\prime \prime}+2 x y^{\prime}=\frac{1}{1+x^{2}}, \quad \lim _{x \rightarrow+\infty} y=\frac{\pi^{2}}{8},\left.\quad y^{\prime}\right|_{x=0}=0$.
670. $(1-x) y^{\prime \prime}+x y^{\prime}-y=(x-1)^{2} e^{2}, \lim _{x \rightarrow-\infty} y=0,\left.y\right|_{x=0}=1 ; \quad y_{1}=x, y_{2}=e^{x}$.
671. $2 x^{2}(2-\ln x) y^{\prime \prime}+x(4-\ln x) y^{\prime}-y=\frac{(2-\ln x)^{2}}{\sqrt{x}}, \quad \lim _{x \rightarrow+\infty} y=0$; $y_{1}=\ln x, y_{2}=\sqrt{x}$.
672. $y^{\prime \prime}+\frac{2}{x} y^{\prime}-y=4 e^{x}, \quad \lim _{x \rightarrow-\infty} y=0,\left.\quad y^{\prime}\right|_{x=-1}=-\frac{1}{e} ; \quad y_{1}=\frac{e^{x}}{x}, \quad y_{2}=\frac{e^{-x}}{x}$.
673. $x^{3}(\ln x-1) y^{\prime \prime}-x^{2} y^{\prime}+x y=2 \ln x, \quad \lim _{x \rightarrow+\infty} y=0 ; \quad y_{1}=x, \quad y_{2}=\ln x$.
674. $\left(x^{2}-2 x