Example 4 (Decomposition of a Substance). Substance $A$ decomposes into two substances $X$ and $Y$ with the rate of formation of each being proportional to the amount of undecomposed substance. Find the law of change of the quantities $x$ and $y$ of substances $X$ and $Y$ as functions of time $t$, if at $t=0$ we have $x=y=0$, and after one hour $x=\frac{a}{8}, y=\frac{3 a}{8}$, where $a$ is the initial amount of substance $A$.
Solution. At time $t$, the amount of undecomposed substance $A$ is $a-x-y$. By the condition of the problem, we have

$$
\left\{\begin{array}{l}
\frac{d x}{d t}=k_{1}(a-x-y) \\
\frac{d y}{d t}=k_{2}(a-x-y)
\end{array}\right.
$$

Dividing the second equation by the first, we get

$$
\frac{d y}{d t}=\frac{k_{2}}{k_{1}}, \quad \text { hence } \quad y=\frac{k_{2}}{k_{1}} x+C_{1}
$$

At $t=0$, we have $x=y=0$, so from the last equation we find $C_{1}=0$, which means

$$
y=\frac{k_{2}}{k_{1}} x
$$

Substituting (9) into the first equation of the system, we get the equation

$$
\frac{d x}{d t}+\left(k_{1}+k_{2}\right) x=k_{1} a
$$

the general solution of which is

$$
x=\frac{k_{1} a}{k_{1}+k_{2}}+C_{2} e^{-\left(k_{1}+k_{2}\right) t}
$$

Using the initial condition $\left.x\right|_{t=0}=0$, we find $C_{2}=-\frac{k_{1} a}{k_{1}+k_{2}}$, so

$$
x=\frac{k_{1} a}{k_{1}+k_{2}}\left[1-e^{-\left(k_{1}+k_{2}\right) t}\right]
$$

Substituting (10) into (9), we get

$$
y=\frac{k_{2} a}{k_{1}+k_{2}}\left[1-e^{-\left(k_{1}+k_{2}\right) t}\right]
$$

To determine the coefficients $k_{1}$ and $k_{2}$, we take one hour as the unit of time. Considering that $x=\frac{a}{8}, y=\frac{3 a}{8}$ at $t=1$, from (10) and (10') we find

$$
\frac{k_{1}}{k_{1}+k_{2}}\left[1-e^{-\left(k_{1}+k_{2}\right)}\right]=\frac{1}{8}, \quad \frac{k_{2}}{k_{1}+k_{2}}\left[1-e^{-\left(k_{1}+k_{2}\right)}\right]=\frac{3}{8}
$$

from which

$$
k_{2}=3 k_{1}, \quad k_{1}+k_{2}=\ln 2
$$

so $k_{1}=\frac{\ln 2}{4}, k_{2}=\frac{3}{4} \ln 2$, and the desired solution of the system (8) is

$$
x=\frac{a}{4}\left(1-2^{-t}\right), \quad y=\frac{3 a}{4}\left(1-2^{-t}\right)
$$