60.3.314 problem 1320

Internal problem ID [11324]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1320
Date solved : Monday, January 27, 2025 at 11:13:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x^{2}-2\right ) y^{\prime \prime }-\left (x^{3}+3 x^{2}-2 x -2\right ) y^{\prime }+\left (x^{2}+4 x +2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 17

dsolve(x*(x^2-2)*diff(diff(y(x),x),x)-(x^3+3*x^2-2*x-2)*diff(y(x),x)+(x^2+4*x+2)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \left (x -1\right )+c_{2} {\mathrm e}^{x} x^{2} \]

✓ Solution by Mathematica

Time used: 0.340 (sec). Leaf size: 126

DSolve[(2 + 4*x + x^2)*y[x] - (-2 - 2*x + 3*x^2 + x^3)*D[y[x],x] + x*(-2 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\left (-\frac {K[1]}{K[1]^2-2}+\frac {1}{2}+\frac {3}{2 K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\left (-\frac {2 K[2]}{K[2]^2-2}-1-\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]^3+K[1]^2-2 K[1]-6}{2 K[1] \left (K[1]^2-2\right )}dK[1]\right )dK[3]+c_1\right ) \]