73.9.1 problem 14.1 (a)

Internal problem ID [15191]
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 : Monday, March 31, 2025 at 01:30:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x^{2} y^{\prime }-4 y&=x^{3} \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 127

ode:=diff(diff(y(x),x),x)+x^2*diff(y(x),x)-4*y(x) = x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -{\mathrm e}^{-\frac {x^{3}}{3}} \left (\int \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) \int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right )^{2}}d x x^{3}d x +\left (-c_1 -\int \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) x^{3}d x \right ) \int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right )^{2}}d x -c_2 \right ) \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) \]

✓ Mathematica. Time used: 0.545 (sec). Leaf size: 194

ode=D[y[x],{x,2}]+x^2*D[y[x],x]-4*y[x]==x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-\frac {x^3}{3}} \text {HeunT}[4,-2,0,0,-1,x] \left (\int _1^x-\frac {e^{\frac {K[2]^3}{3}} \text {HeunT}[4,0,0,0,1,K[2]] K[2]^3}{\text {HeunT}[4,-2,0,0,-1,K[2]] \text {HeunTPrime}[4,0,0,0,1,K[2]]+\text {HeunT}[4,0,0,0,1,K[2]] \left (\text {HeunT}[4,-2,0,0,-1,K[2]] K[2]^2-\text {HeunTPrime}[4,-2,0,0,-1,K[2]]\right )}dK[2]+c_2\right )+\text {HeunT}[4,0,0,0,1,x] \left (\int _1^x\frac {\text {HeunT}[4,-2,0,0,-1,K[1]] K[1]^3}{\text {HeunT}[4,-2,0,0,-1,K[1]] \text {HeunTPrime}[4,0,0,0,1,K[1]]+\text {HeunT}[4,0,0,0,1,K[1]] \left (\text {HeunT}[4,-2,0,0,-1,K[1]] K[1]^2-\text {HeunTPrime}[4,-2,0,0,-1,K[1]]\right )}dK[1]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 + x**2*Derivative(y(x), x) - 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3 + 4*y(x) - Derivative(y(x), (x, 2)))/x**2 cannot be solved by the factorable group method