Internal problem ID [8644]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1064.
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 }+\tan \relax (x )+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 138
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+tan(x)+b*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\left (-\frac {a}{2}+\frac {\sqrt {a^{2}-4 b}}{2}\right ) x} c_{2}+{\mathrm e}^{\left (-\frac {a}{2}-\frac {\sqrt {a^{2}-4 b}}{2}\right ) x} c_{1}+\frac {\left (\left (\int \tan \relax (x ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{\frac {\left (a -\sqrt {a^{2}-4 b}\right ) x}{2}}-\left (\int \tan \relax (x ) {\mathrm e}^{\frac {\left (a -\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}\right ) {\mathrm e}^{-a x}}{\sqrt {a^{2}-4 b}} \]
✓ Solution by Mathematica
Time used: 0.323 (sec). Leaf size: 485
DSolve[Tan[x] + b*y[x] + a*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (b \left (i \sqrt {a^2-4 b}-i a+4\right ) e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a+4 i\right )} \, _2F_1\left (1,\frac {1}{4} \left (-i a-i \sqrt {a^2-4 b}+4\right );\frac {1}{4} \left (-i a-i \sqrt {a^2-4 b}+8\right );-e^{2 i x}\right )+i b \left (\sqrt {a^2-4 b}+a+4 i\right ) e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a+4 i\right )} \, _2F_1\left (1,\frac {1}{4} \left (-i a+i \sqrt {a^2-4 b}+4\right );\frac {1}{4} \left (-i a+i \sqrt {a^2-4 b}+8\right );-e^{2 i x}\right )+(2 a-i b+4 i) \left (e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (\left (\sqrt {a^2-4 b}+a\right ) \, _2F_1\left (1,\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right );\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right )+1;-e^{2 i x}\right )+\left (\sqrt {a^2-4 b}-a\right ) \, _2F_1\left (1,-\frac {1}{4} i \left (a+\sqrt {a^2-4 b}\right );\frac {1}{4} \left (-i a-i \sqrt {a^2-4 b}+4\right );-e^{2 i x}\right )\right )+2 i b \sqrt {a^2-4 b} \left (c_2 e^{x \sqrt {a^2-4 b}}+c_1\right )\right )\right )}{2 b \sqrt {a^2-4 b} (2 i a+b-4)} \\ \end{align*}