15. Knowing that the function $y=C x+1$ is the general solution of the equation $x y^{\prime}=y-1$, determine its particular solution if $y(1)=5$.
Solution. Substituting the given initial conditions \(x=1, y=5\) into the general solution \(y=Cx+1\), we get \(5=C \cdot 1+1\), from which \(C=4\).

Now, substituting the value \(C=4\) into the general solution, we find the particular solution \(y=4x+1\).

Thus, when solving a differential equation, the general solution is obtained first. Then, if initial conditions are known, a particular solution can be obtained. For this, you need to:

1) substitute the initial conditions into the general solution and calculate \(C\);
2) substitute the numerical value of \(C\) into the general solution.

The problem of finding a specific particular solution of a given differential equation from initial conditions is called the Cauchy problem.