23.3.72 problem 74
Internal
problem
ID
[5786]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
3.
THE
DIFFERENTIAL
EQUATION
IS
LINEAR
AND
OF
SECOND
ORDER,
page
311
Problem
number
:
74
Date
solved
:
Tuesday, September 30, 2025 at 02:03:12 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} \csc \left (a \right )^{2} y-2 \tan \left (a \right ) y^{\prime }+y^{\prime \prime }&={\mathrm e}^{x \tan \left (a \right )} x^{2} \end{align*}
✓ Maple. Time used: 0.130 (sec). Leaf size: 133
ode:=csc(a)^2*y(x)-2*tan(a)*diff(y(x),x)+diff(diff(y(x),x),x) = exp(x*tan(a))*x^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {{\mathrm e}^{x \tan \left (a \right )} \left (c_2 \left (\tan \left (a \right )^{4}-2 \sec \left (a \right )^{2}+\csc \left (a \right )^{4}\right ) {\mathrm e}^{-\csc \left (a \right ) \sec \left (a \right )^{2} \sqrt {\cos \left (a \right )^{6}-3 \cos \left (a \right )^{4}+\cos \left (a \right )^{2}}\, x}+c_1 \left (\tan \left (a \right )^{4}-2 \sec \left (a \right )^{2}+\csc \left (a \right )^{4}\right ) {\mathrm e}^{\csc \left (a \right ) \sec \left (a \right )^{2} \sqrt {\cos \left (a \right )^{6}-3 \cos \left (a \right )^{4}+\cos \left (a \right )^{2}}\, x}+\csc \left (a \right )^{2} x^{2}-\tan \left (a \right )^{2} x^{2}-2\right )}{\tan \left (a \right )^{4}-2 \sec \left (a \right )^{2}+\csc \left (a \right )^{4}}
\]
✓ Mathematica. Time used: 0.255 (sec). Leaf size: 97
ode=Csc[a]^2*y[x] - 2*Tan[a]*D[y[x],x] + D[y[x],{x,2}] == E^(x*Tan[a])*x^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {e^{x \tan (a)} \left (-x^2 \tan ^2(a)+x^2 \csc ^2(a)-2\right )}{\left (\csc ^2(a)-\tan ^2(a)\right )^2}+c_1 e^{x \left (\tan (a)-\sqrt {\tan ^2(a)-\csc ^2(a)}\right )}+c_2 e^{x \left (\tan (a)+\sqrt {\tan ^2(a)-\csc ^2(a)}\right )} \end{align*}
✓ Sympy. Time used: 0.485 (sec). Leaf size: 134
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-x**2*exp(x*tan(a)) + y(x)/sin(a)**2 - 2*tan(a)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{x \left (- \frac {\sqrt {\left (\frac {\sin ^{4}{\left (a \right )}}{\cos ^{2}{\left (a \right )}} - 1\right ) \sin ^{2}{\left (a \right )}}}{\sin ^{2}{\left (a \right )}} + \tan {\left (a \right )}\right )} + C_{2} e^{x \left (\frac {\sqrt {\left (\frac {\sin ^{4}{\left (a \right )}}{\cos ^{2}{\left (a \right )}} - 1\right ) \sin ^{2}{\left (a \right )}}}{\sin ^{2}{\left (a \right )}} + \tan {\left (a \right )}\right )} - \frac {x^{2} e^{x \tan {\left (a \right )}} \sin ^{2}{\left (a \right )}}{\sin ^{2}{\left (a \right )} \tan ^{2}{\left (a \right )} - 1} - \frac {2 e^{x \tan {\left (a \right )}} \sin ^{4}{\left (a \right )}}{\sin ^{4}{\left (a \right )} \tan ^{4}{\left (a \right )} - 2 \sin ^{2}{\left (a \right )} \tan ^{2}{\left (a \right )} + 1}
\]