60.3.342 problem 1348

Internal problem ID [11352]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1348
Date solved : Tuesday, January 28, 2025 at 06:04:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 1.320 (sec). Leaf size: 73

dsolve(diff(diff(y(x),x),x) = -1/x*diff(y(x),x)-(b*x^2+a*(x^4+1))/x^4*y(x),y(x), singsol=all)
 

\[ y = \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right ) \left (c_{1} +c_{2} \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right )^{2}}d x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.490 (sec). Leaf size: 34

DSolve[D[y[x],{x,2}] == -(((b*x^2 + a*(1 + x^4))*y[x])/x^4) - D[y[x],x]/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \text {MathieuC}[-b,a,i \log (x)]+c_2 \text {MathieuS}[-b,a,i \log (x)] \]