76.48.4 problem Ex. 4

Internal problem ID [20270]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. End of chapter problems at page 107
Problem number : Ex. 4
Date solved : Thursday, October 02, 2025 at 05:40:42 PM
CAS classification : [[_3rd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{3} y^{\prime \prime \prime }+4 x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+3 x^{2} y&=2 x \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

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

\[ y = \operatorname {BesselJ}\left (0, x\right ) c_3 +\operatorname {BesselY}\left (0, x\right ) c_2 +\frac {\pi \operatorname {StruveH}\left (0, x\right )}{2}+\operatorname {LommelS1}\left (-2, 0, x\right ) c_1 \]

✓ Mathematica. Time used: 0.811 (sec). Leaf size: 364

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

\begin{align*} y(x)&\to \frac {2 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right ) \left (\int _1^x-\frac {9 \pi (\operatorname {BesselJ}(1,K[3]) \operatorname {BesselY}(0,K[3])-\operatorname {BesselJ}(0,K[3]) \operatorname {BesselY}(1,K[3])) K[3]^2}{9 \left (K[3]^2+1\right ) (\pi K[3] \pmb {H}_0(K[3])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4}dK[3]+c_3\right )}{x}+\operatorname {BesselJ}(0,x) \int _1^x\frac {18 \pi \left (2 \operatorname {BesselY}(0,K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) (\operatorname {BesselY}(0,K[1])-\operatorname {BesselY}(1,K[1]) K[1])\right )}{9 \left (K[1]^2+1\right ) (\pi K[1] \pmb {H}_0(K[1])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^4}dK[1]+2 \operatorname {BesselY}(0,x) \int _1^x-\frac {9 \pi \left (2 \operatorname {BesselJ}(0,K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) (\operatorname {BesselJ}(0,K[2])-\operatorname {BesselJ}(1,K[2]) K[2])\right )}{9 \left (K[2]^2+1\right ) (\pi K[2] \pmb {H}_0(K[2])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4}dK[2]+c_1 \operatorname {BesselJ}(0,x)+2 c_2 \operatorname {BesselY}(0,x) \end{align*}

✗ Sympy

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

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