60.3.59 problem 1064
Internal
problem
ID
[11055]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1064
Date
solved
:
Sunday, March 30, 2025 at 07:40:32 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+a y^{\prime }+\tan \left (x \right )+b y&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 113
ode:=diff(diff(y(x),x),x)+a*diff(y(x),x)+tan(x)+b*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {{\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}} \left (\left (-\sqrt {a^{2}-4 b}\, c_2 +\int \tan \left (x \right ) {\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{x \sqrt {a^{2}-4 b}}-c_1 \sqrt {a^{2}-4 b}-\int \tan \left (x \right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right )}{\sqrt {a^{2}-4 b}}
\]
✓ Mathematica. Time used: 0.556 (sec). Leaf size: 801
ode=Tan[x] + b*y[x] + a*D[y[x],x] + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {2 e^{-\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) x} \left (-c_1 a^3-e^{\sqrt {a^2-4 b} x} c_2 a^3+\sqrt {a^2-4 b} c_1 a^2-2 i c_1 a^2+\sqrt {a^2-4 b} e^{\sqrt {a^2-4 b} x} c_2 a^2-2 i e^{\sqrt {a^2-4 b} x} c_2 a^2+4 b c_1 a+2 i \sqrt {a^2-4 b} c_1 a+4 b e^{\sqrt {a^2-4 b} x} c_2 a+2 i \sqrt {a^2-4 b} e^{\sqrt {a^2-4 b} x} c_2 a-i e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right ),\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right )+1,-e^{2 i x}\right ) a-2 \sqrt {a^2-4 b} b c_1+8 i b c_1-2 \sqrt {a^2-4 b} b e^{\sqrt {a^2-4 b} x} c_2+8 i b e^{\sqrt {a^2-4 b} x} c_2-i \left (\sqrt {a^2-4 b}-a\right ) e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}+4 i\right ) x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i a}{4}+\frac {1}{4} i \sqrt {a^2-4 b}+1,-\frac {i a}{4}+\frac {1}{4} i \sqrt {a^2-4 b}+2,-e^{2 i x}\right )+i \sqrt {a^2-4 b} e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right ),\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right )+1,-e^{2 i x}\right )+4 e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right ),\frac {1}{4} i \left (\sqrt {a^2-4 b}-a\right )+1,-e^{2 i x}\right )+\left (-a^3+\left (\sqrt {a^2-4 b}-2 i\right ) a^2+\left (4 b+2 i \sqrt {a^2-4 b}\right ) a-2 \left (\sqrt {a^2-4 b}-4 i\right ) b\right ) \int _1^x\frac {e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) K[1]} \tan (K[1])}{\sqrt {a^2-4 b}}dK[1]\right )}{\left (\sqrt {a^2-4 b}-a\right ) \left (-a+\sqrt {a^2-4 b}-4 i\right ) \sqrt {a^2-4 b}}
\]
✓ Sympy. Time used: 14.852 (sec). Leaf size: 148
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*Derivative(y(x), x) + b*y(x) + tan(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {x \left (- a + \sqrt {a^{2} - 4 b}\right )}{2}} + C_{2} e^{- \frac {x \left (a + \sqrt {a^{2} - 4 b}\right )}{2}} - \frac {e^{\frac {x \left (- a + \sqrt {a^{2} - 4 b}\right )}{2}} \int e^{\frac {a x}{2}} e^{- \frac {x \sqrt {a^{2} - 4 b}}{2}} \tan {\left (x \right )}\, dx}{\sqrt {a^{2} - 4 b}} + \frac {e^{- \frac {x \left (a + \sqrt {a^{2} - 4 b}\right )}{2}} \int e^{\frac {a x}{2}} e^{\frac {x \sqrt {a^{2} - 4 b}}{2}} \tan {\left (x \right )}\, dx}{\sqrt {a^{2} - 4 b}}
\]