60.3.265 problem 1270

Internal problem ID [11275]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1270
Date solved : Monday, January 27, 2025 at 11:04:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+6 x +4\right ) y^{\prime \prime }+\left (10 x^{2}+21 x +8\right ) y^{\prime }+\left (12 x^{2}+17 x +8\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.923 (sec). Leaf size: 46

dsolve((2*x^2+6*x+4)*diff(diff(y(x),x),x)+(10*x^2+21*x+8)*diff(y(x),x)+(12*x^2+17*x+8)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-2 x} \left (\left (x +1\right )^{{5}/{2}} c_{2} \operatorname {HeunC}\left (-1, \frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right )+c_{1} \operatorname {HeunC}\left (-1, -\frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right )\right ) \left (x +2\right )^{4} \]

✓ Solution by Mathematica

Time used: 0.337 (sec). Leaf size: 120

DSolve[(8 + 17*x + 12*x^2)*y[x] + (8 + 21*x + 10*x^2)*D[y[x],x] + (4 + 6*x + 2*x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]-2 K[1]^2}{4 K[1]^2+12 K[1]+8}dK[1]-\frac {1}{2} \int _1^x\left (-\frac {3}{K[2]+2}+5-\frac {3}{2 (K[2]+1)}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2 K[1]^2}{4 K[1]^2+12 K[1]+8}dK[1]\right )dK[3]+c_1\right ) \]