Example 6. Find the solution of the equation $y^{\prime}-y=\cos x-\sin x$, satisfying the condition: $y$ is bounded as $x \rightarrow+\infty$.
Solution. The general solution of the given equation is

$$
y=C e^{x}+\sin x
$$

Any solution of the equation obtained from the general solution for $C \neq 0$ will be unbounded, as when $x \rightarrow+\infty$ the function $\sin x$ is bounded, while $e^{x} \rightarrow+\infty$. Therefore, the equation has a unique bounded solution $y=\sin x$ as $x \rightarrow+\infty$, which is obtained from the general solution when $C=0$.

## Problems for Independent Solution

Solve the following linear equations. Solve the Cauchy problem where indicated:
125. $y^{\prime}+2 y=e^{-x}$.
126. $x^{2}+x y^{\prime}=y,\left.y\right|_{x=1}=0$.
127. $y^{\prime}-2 x y=2 x e^{x^{2}}$
128. $y^{\prime}+2 x y=e^{-x^{2}}$.
129. $y^{\prime} \cos x-y \sin x=2 x,\left.y\right|_{x=0}=0$.
130. $x y^{\prime}-2 y=x^{3} \cos x$.
131. $y^{\prime}-y \operatorname{tg} x=\frac{1}{\cos ^{3} x},\left.y\right|_{x=0}=0$.
132. $y^{\prime} x \ln x-y=3 x^{3} \ln ^{2} x$.
133. $\left(2 x-y^{2}\right) y^{\prime}=2 y$.
134. $y^{\prime}+y \cos x=\cos x,\left.y\right|_{x=0}=1$.
135. $y^{\prime}=\frac{y}{2 y \ln y+y-x}$.
136. $\left(e^{-y^{2} / 2}-x y\right) d y-d x=0$.
137. $y^{\prime}-y e^{x}=2 x e^{x}$.
138. $y^{\prime}+x e^{x} y=e^{(1-x) e^{x}}$.

139. Find the current $i(t)$ given that $E(t)=E_{0} \sin 2 \pi n t, i(0)=I_{0}$, where $E_{0}$, $I_{0}=$ const.
140. A capacitor with capacitance $Q$ is connected to a circuit with voltage $E$ and resistance $R$. Determine the charge $q$ on the capacitor at time $t$ after connection.
141. A point of mass $m$ moves in a straight line under the action of a force proportional to time (proportionality coefficient $k_{1}$). Additionally, the point experiences resistance proportional to its velocity (proportionality coefficient $k_{2}$). Find the dependence of velocity on time, assuming that the initial velocity is zero.
142. Find the curves that have the property that the segment cut off by the tangent at any point on the curve on the $O y$ axis is equal to the square of the abscissa of the point of tangency.
143. Find the curve for which the segment cut off by the tangent on the ordinate axis is equal to half the sum of the coordinates of the point of tangency.
144. Find the general solution of the linear inhomogeneous equation of the first order $y^{\prime}+p(x) y=q(x)$ if one particular solution $y_{1}(x)$ is known.
145. Find the general solution of the linear inhomogeneous equation of the first order $y^{\prime}+p(x) y=q(x)$ if two particular solutions $y_{1}(x)$ and $y_{2}(x)$ are known.
146. Show that a linear equation remains linear under any change of the independent variable $x=\varphi(t)$, where $\varphi(t)$ is a differentiable function.
147. Show that a linear equation remains linear under any linear transformation of the unknown function $y=\alpha(x) z+\beta(x)$, where $\alpha(x)$ and $\beta(x)$ are arbitrary differentiable functions, and $\alpha(x) \neq 0$ in the considered interval.

In the following problems, find the solutions of the equations that satisfy the given conditions:

148. $y^{\prime}-y \ln 2=2^{\sin x}(\cos x-1) \ln 2, y$ is bounded as $x \rightarrow+\infty$.
149. $y^{\prime}-y=-2 e^{-x}, y \rightarrow 0$ as $x \rightarrow+\infty$.
150. $y^{\prime} \sin x-y \cos x=-\frac{\sin ^{2} x}{x^{2}}, y \rightarrow 0$ as $x \rightarrow \infty$.
151. $x^{2} y^{\prime} \cos \frac{1}{x}-y \sin \frac{1}{x}=-1, y \rightarrow 1$ as $x \rightarrow \infty$.
152. $2 x y^{\prime}-y=1-\frac{2}{\sqrt{x}}, y \rightarrow-1$ as $x \rightarrow+\infty$.
153. $x^{2} y^{\prime}+y=\left(x^{2}+1\right) e^{x}, y \rightarrow 1$ as $x \rightarrow-\infty$.
154. $x y^{\prime}+y=2 x, y$ is bounded as $x \rightarrow 0$.
155. $y^{\prime} \sin x+y \cos x=1, y$ is bounded as $x \rightarrow 0$.
156. $y^{\prime} \cos x-y \sin x=-\sin 2 x, y \rightarrow 0$ as $x \rightarrow \frac{\pi}{2}$.

$2^{\circ}$. Bernoulli Equation. The Bernoulli equation is of the form

$$
\frac{d y}{d x}+p(x) y=q(x) y^{n}
$$

where $n \neq 0,1$ (for $n=0$ and $n=1$ the equation is linear).

By using the substitution $z=\frac{1}{y^{n-1}}$, the Bernoulli equation can be reduced to a linear equation and solved as a linear equation.