3.43. Solve the Cauchy problem:

$$
\left\{\begin{array}{l}
y^{\prime}(t)+a y(t)=\varphi(t) \\
y(0)=y_{0} \\
t>0
\end{array}\right.
$$
Solution. Transitioning to images, we get:

$$
p \bar{y}(p)-y_{0}+a \bar{y}(p)=\bar{\varphi}(p)
$$

or

$$
\bar{y}(p)=\frac{\bar{\varphi}(p)+y_{0}}{p+a}
$$

from which

$$
\bar{y}(p)=\frac{y_{0}}{p+a}+\frac{\bar{\varphi}(p)}{p+a}
$$

Transitioning to originals, using Table 3.2 and the convolution theorem, we have:

$$
y(t)=y_{0} e^{-a t}+\int_{0}^{t} \varphi(\tau) e^{-a(t-\tau)} d \tau
$$