60.4.45 problem 1493
Internal
problem
ID
[11496]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
3,
linear
third
order
Problem
number
:
1493
Date
solved
:
Tuesday, January 28, 2025 at 06:06:38 PM
CAS
classification
:
[[_3rd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 y x -f \left (x \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 1035
dsolve(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)-f(x)=0,y(x), singsol=all)
\[
\text {Expression too large to display}
\]
✓ Solution by Mathematica
Time used: 1.048 (sec). Leaf size: 373
DSolve[-f[x] + 3*x*y[x] + (2 + x^2)*D[y[x],x] + 4*x*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {2 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right ) \left (\int _1^x\frac {9 \pi (\operatorname {BesselJ}(1,K[3]) \operatorname {BesselY}(0,K[3])-\operatorname {BesselJ}(0,K[3]) \operatorname {BesselY}(1,K[3])) f(K[3]) K[3]^2}{32 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4-18 \left (K[3]^2+1\right ) (\pi K[3] \pmb {H}_0(K[3])-2)}dK[3]+c_3\right )}{x}+\operatorname {BesselJ}(0,x) \int _1^x\frac {9 \pi f(K[1]) \left (2 \operatorname {BesselY}(0,K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) (\operatorname {BesselY}(0,K[1])-\operatorname {BesselY}(1,K[1]) K[1])\right )}{9 \left (K[1]^2+1\right ) (\pi K[1] \pmb {H}_0(K[1])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^4}dK[1]+2 \operatorname {BesselY}(0,x) \int _1^x\frac {9 \pi f(K[2]) \left (2 \operatorname {BesselJ}(0,K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) (\operatorname {BesselJ}(0,K[2])-\operatorname {BesselJ}(1,K[2]) K[2])\right )}{32 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4-18 \left (K[2]^2+1\right ) (\pi K[2] \pmb {H}_0(K[2])-2)}dK[2]+c_1 \operatorname {BesselJ}(0,x)+2 c_2 \operatorname {BesselY}(0,x)
\]