82.48.4 problem Ex. 4
Internal
problem
ID
[18985]
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
:
Tuesday, January 28, 2025 at 12:43:52 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*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 27
dsolve(x^3*diff(y(x),x$3)+4*x^2*diff(y(x),x$2)+x*(x^2+2)*diff(y(x),x)+3*x^2*y(x)=2*x,y(x), singsol=all)
\[
y \left (x \right ) = \operatorname {BesselJ}\left (0, x\right ) c_3 +\operatorname {BesselY}\left (0, x\right ) c_{2} +\frac {\pi \operatorname {StruveH}\left (0, x\right )}{2}+\operatorname {LommelS1}\left (-2, 0, x\right ) c_{1}
\]
✓ Solution by Mathematica
Time used: 1.040 (sec). Leaf size: 364
DSolve[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,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {2 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right ) \left (\int _1^x-\frac {9 \pi (\operatorname {BesselJ}(1,K[3]) \operatorname {BesselY}(0,K[3])-\operatorname {BesselJ}(0,K[3]) \operatorname {BesselY}(1,K[3])) K[3]^2}{9 \left (K[3]^2+1\right ) (\pi K[3] \pmb {H}_0(K[3])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4}dK[3]+c_3\right )}{x}+\operatorname {BesselJ}(0,x) \int _1^x\frac {18 \pi \left (2 \operatorname {BesselY}(0,K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) (\operatorname {BesselY}(0,K[1])-\operatorname {BesselY}(1,K[1]) K[1])\right )}{9 \left (K[1]^2+1\right ) (\pi K[1] \pmb {H}_0(K[1])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^4}dK[1]+2 \operatorname {BesselY}(0,x) \int _1^x-\frac {9 \pi \left (2 \operatorname {BesselJ}(0,K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) (\operatorname {BesselJ}(0,K[2])-\operatorname {BesselJ}(1,K[2]) K[2])\right )}{9 \left (K[2]^2+1\right ) (\pi K[2] \pmb {H}_0(K[2])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4}dK[2]+c_1 \operatorname {BesselJ}(0,x)+2 c_2 \operatorname {BesselY}(0,x)
\]