60.4.56 problem 1504

Internal problem ID [11507]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1504
Date solved : Tuesday, January 28, 2025 at 06:06:41 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 18

dsolve((x^2+2)*diff(diff(diff(y(x),x),x),x)-2*x*diff(diff(y(x),x),x)+(x^2+2)*diff(y(x),x)-2*x*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} x^{2}+\cos \left (x \right ) c_{2} +c_3 \sin \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.199 (sec). Leaf size: 212

DSolve[-2*x*y[x] + (2 + x^2)*D[y[x],x] - 2*x*D[y[x],{x,2}] + (2 + x^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} x^2 \left (c_2 \int _1^x\exp \left (\int _1^{K[4]}\left (-\frac {K[1]}{K[1]^2+2}-i\right )dK[1]-\frac {1}{2} \int _1^{K[4]}\frac {4 K[2]^2+12}{K[2]^3+2 K[2]}dK[2]\right ) (K[4]-2 i)dK[4]+c_3 \int _1^x\exp \left (\int _1^{K[5]}\left (-\frac {K[1]}{K[1]^2+2}-i\right )dK[1]-\frac {1}{2} \int _1^{K[5]}\frac {4 K[2]^2+12}{K[2]^3+2 K[2]}dK[2]\right ) (K[5]-2 i) \int _1^{K[5]}\frac {\exp \left (-2 \int _1^{K[3]}\left (-\frac {K[1]}{K[1]^2+2}-i\right )dK[1]\right )}{(K[3]-2 i)^2}dK[3]dK[5]+c_1\right ) \]