Internal problem ID [9399]
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]]
\[ \boxed {y^{\prime \prime }+a y^{\prime }+b y=-\tan \left (x \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 139
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+tan(x)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\left (-\frac {a}{2}+\frac {\sqrt {a^{2}-4 b}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {a}{2}-\frac {\sqrt {a^{2}-4 b}}{2}\right ) x} c_{1} -\frac {{\mathrm e}^{-a x} \left (\left (\int \tan \left (x \right ) {\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}-\left (\int \tan \left (x \right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}\right )}{\sqrt {a^{2}-4 b}} \]
✓ Solution by Mathematica
Time used: 0.752 (sec). Leaf size: 502
DSolve[Tan[x] + b*y[x] + a*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (b \left (i \sqrt {a^2-4 b}-i a+4\right ) e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a+4 i\right )} \operatorname {Hypergeometric2F1}\left (1,-\frac {i a}{4}-\frac {1}{4} i \sqrt {a^2-4 b}+1,-\frac {i a}{4}-\frac {1}{4} i \sqrt {a^2-4 b}+2,-e^{2 i x}\right )+i b \left (\sqrt {a^2-4 b}+a+4 i\right ) e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a+4 i\right )} \operatorname {Hypergeometric2F1}\left (1,-\frac {i a}{4}+\frac {1}{4} i \sqrt {a^2-4 b}+1,-\frac {i a}{4}+\frac {1}{4} i \sqrt {a^2-4 b}+2,-e^{2 i x}\right )+(2 a-i (b-4)) \left (\left (\sqrt {a^2-4 b}+a\right ) e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right ),\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right )+1,-e^{2 i x}\right )+\left (\sqrt {a^2-4 b}-a\right ) e^{\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{4} i \left (a+\sqrt {a^2-4 b}\right ),\frac {1}{4} \left (-i a-i \sqrt {a^2-4 b}+4\right ),-e^{2 i x}\right )+2 i b \sqrt {a^2-4 b} \left (c_2 e^{x \sqrt {a^2-4 b}}+c_1\right )\right )\right )}{2 b \sqrt {a^2-4 b} (2 i a+b-4)} \]