Internal problem ID [8616]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1036.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a y^{\prime }+b y-f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 138
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+b*y(x)-f(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\left (-\frac {a}{2}+\frac {\sqrt {a^{2}-4 b}}{2}\right ) x} c_{2}+{\mathrm e}^{\left (-\frac {a}{2}-\frac {\sqrt {a^{2}-4 b}}{2}\right ) x} c_{1}+\frac {\left (-\left (\int f \relax (x ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{\frac {\left (a -\sqrt {a^{2}-4 b}\right ) x}{2}}+\left (\int f \relax (x ) {\mathrm e}^{\frac {\left (a -\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}\right ) {\mathrm e}^{-a x}}{\sqrt {a^{2}-4 b}} \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 137
DSolve[-f[x] + b*y[x] + a*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (\int _1^x-\frac {e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) K[1]} f(K[1])}{\sqrt {a^2-4 b}}dK[1]+e^{x \sqrt {a^2-4 b}} \left (\int _1^x\frac {e^{\frac {1}{2} \left (a-\sqrt {a^2-4 b}\right ) K[2]} f(K[2])}{\sqrt {a^2-4 b}}dK[2]+c_2\right )+c_1\right ) \\ \end{align*}