73.9.1 problem 14.1 (a)
Internal
problem
ID
[15262]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
14.
Higher
order
equations
and
the
reduction
of
order
method.
Additional
exercises
page
277
Problem
number
:
14.1
(a)
Date
solved
:
Tuesday, January 28, 2025 at 08:25:31 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+x^{2} y^{\prime }-4 y&=x^{3} \end{align*}
✓ Solution by Maple
Time used: 0.878 (sec). Leaf size: 127
dsolve(diff(y(x),x$2)+x^2*diff(y(x),x)-4*y(x)=x^3,y(x), singsol=all)
\[
y = -\left (\int \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right )^{2}}d x \right ) x^{3}d x +\left (-c_{1} -\int \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) x^{3}d x \right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right )^{2}}d x \right )-c_{2} \right ) \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) {\mathrm e}^{-\frac {x^{3}}{3}}
\]
✓ Solution by Mathematica
Time used: 0.545 (sec). Leaf size: 194
DSolve[D[y[x],{x,2}]+x^2*D[y[x],x]-4*y[x]==x^3,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to e^{-\frac {x^3}{3}} \text {HeunT}[4,-2,0,0,-1,x] \left (\int _1^x-\frac {e^{\frac {K[2]^3}{3}} \text {HeunT}[4,0,0,0,1,K[2]] K[2]^3}{\text {HeunT}[4,-2,0,0,-1,K[2]] \text {HeunTPrime}[4,0,0,0,1,K[2]]+\text {HeunT}[4,0,0,0,1,K[2]] \left (\text {HeunT}[4,-2,0,0,-1,K[2]] K[2]^2-\text {HeunTPrime}[4,-2,0,0,-1,K[2]]\right )}dK[2]+c_2\right )+\text {HeunT}[4,0,0,0,1,x] \left (\int _1^x\frac {\text {HeunT}[4,-2,0,0,-1,K[1]] K[1]^3}{\text {HeunT}[4,-2,0,0,-1,K[1]] \text {HeunTPrime}[4,0,0,0,1,K[1]]+\text {HeunT}[4,0,0,0,1,K[1]] \left (\text {HeunT}[4,-2,0,0,-1,K[1]] K[1]^2-\text {HeunTPrime}[4,-2,0,0,-1,K[1]]\right )}dK[1]+c_1\right )
\]