23.5.81 problem 81

Internal problem ID [6690]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 81
Date solved : Friday, October 03, 2025 at 02:09:44 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 1037

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 0.666 (sec). Leaf size: 373

ode=3*x*y[x] + (2 + x^2)*D[y[x],x] + 4*x*D[y[x],{x,2}] + x^2*D[y[x],{x,3}] == f[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])) f(K[3]) K[3]^2}{32 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4-18 \left (K[3]^2+1\right ) (\pi K[3] \pmb {H}_0(K[3])-2)}dK[3]+c_3\right )}{x}+\operatorname {BesselJ}(0,x) \int _1^x\frac {9 \pi f(K[1]) \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 f(K[2]) \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 )}{32 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4-18 \left (K[2]^2+1\right ) (\pi K[2] \pmb {H}_0(K[2])-2)}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**2*Derivative(y(x), (x, 3)) + 3*x*y(x) + 4*x*Derivative(y(x), (x, 2)) + (x**2 + 2)*Derivative(y(x), x) - f(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)