General form of a linear differential equation of order one

\[y' + ya(x) = s(x)\]

The equation is called homogeneous when \(s(x) = 0\) otherwise it is called inhomogeneous.

To solve such type of equation we need \(y_h(x)\) and \(y_p(x)\).

\[y(x) = y_h(x) + y_p(x)\]


Where \(y_h(x)\) is the solution of the homogeneous linear differential equation with the form

\[y' + ya(x) = 0\]

To get \(y_h(x)\) we have to separate x and y and calculate the integral for both sides

\[\frac{dy}{dx} + ya(x) = 0\] \[dy+ya(x)dx = 0\] \[\frac{dy}{y} + a(x)dx = 0\]

The next step is called separation of variables

\[\frac{dy}{y} = -a(x)dx\] \[\int \frac{1}{y}dy = - \int a(x)dx\] \[ln|y| = - \int a(x)dx + C\] \[y = C * e^{- \int a(x)dx}\]

for \(C = \pm e^{C_0}\), \(C_0 \epsilon \R\) and wee need \(C = 0\) for \(y = 0\), because \(e^x \ne 0\).

Now we can solve \(y_h(x)\)

\[y_h(x) = C * e^{- \int a(x)dx}\]


To get a particular solution \(y_p(x)\) we have to change our constant \(C\) to a function \(C(x)\). This step is called variation of constants.

\[y_p(x) = C(x) * e^{- \int a(x)dx}\]

In the original equation we can replace \(y'\) with \(y_p'(x)\) and \(y\) with \(y_p(x)\).

\[y_p'(x) + y_p(x)a(x) = s(x)\] \[C'(x) * (e^{- \int a(x)dx}) + C(x) * e^{- \int a(x)dx}*a(x) = s(x)\] \[C'(x) * e^{- \int a(x)dx} - C(x)*e^{- \int a(x)dx}*a(x) + C(x) * e^{- \int a(x)dx}*a(x) = s(x)\] \[C'(x) = s(x)e^{\int a(x)dx}\] \[C(x) = \int s(x)e^{\int a(x)dx}dx\]


Now we can solve our equation

\[y(x) = y_h(x) + y_p(x)\] \[y(x) = C * e^{- \int a(x)dx} + C(x) * e^{- \int a(x)dx}\]