77.27.14 problem 14
Internal
problem
ID
[20645]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
VII.
Exact
differential
equations
and
certain
particular
forms
of
equations.
Exercise
VII
(A)
at
page
104
Problem
number
:
14
Date
solved
:
Friday, October 03, 2025 at 07:45:54 AM
CAS
classification
:
[[_3rd_order, _linear, _nonhomogeneous]]
\begin{align*} 3 y x +y^{\prime } \left (x^{2}+2\right )+4 x y^{\prime \prime }+x^{2} y^{\prime \prime \prime }&=2 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 48
ode:=x^2*diff(diff(diff(y(x),x),x),x)+4*x*diff(diff(y(x),x),x)+(x^2+2)*diff(y(x),x)+3*x*y(x) = 2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {c_1 \operatorname {BesselJ}\left (0, x\right ) x +c_3 \operatorname {MeijerG}\left (\left [\left [-\frac {1}{2}\right ], \left [\right ]\right ], \left [\left [0, 0, -\frac {1}{2}\right ], \left [\right ]\right ], \frac {x^{2}}{4}\right ) x +c_2 \operatorname {hypergeom}\left (\left [1\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -\frac {x^{2}}{4}\right )+1}{x}
\]
✓ Mathematica. Time used: 0.451 (sec). Leaf size: 364
ode=x^2*D[y[x],{x,3}]+4*x*D[y[x],{x,2}]+(x^2+2)*D[y[x],x]+3*x*y[x]==2;
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**2*Derivative(y(x), (x, 3)) + 3*x*y(x) + 4*x*Derivative(y(x), (x, 2)) + (x**2 + 2)*Derivative(y(x), x) - 2,0)
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