60.3.343 problem 1349

Internal problem ID [11353]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1349
Date solved : Tuesday, January 28, 2025 at 06:04:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.150 (sec). Leaf size: 66

dsolve(diff(diff(y(x),x),x) = -(x^2+1)/x^3*diff(y(x),x)-1/x^4*y(x),y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\frac {1}{4 x^{2}}} \left (c_{1} \left (2 x^{2}-1\right ) \operatorname {BesselI}\left (0, -\frac {1}{4 x^{2}}\right )+\left (2 x^{2}-1\right ) c_{2} \operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right )-\operatorname {BesselI}\left (1, -\frac {1}{4 x^{2}}\right ) c_{1} +\operatorname {BesselK}\left (1, -\frac {1}{4 x^{2}}\right ) c_{2} \right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.216 (sec). Leaf size: 73

DSolve[D[y[x],{x,2}] == -(y[x]/x^4) - ((1 + x^2)*D[y[x],x])/x^3,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| \begin {array}{c} \frac {3}{2} \\ 0,0 \\ \end {array} \right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) \operatorname {BesselI}\left (0,\frac {1}{4 x^2}\right )+\operatorname {BesselI}\left (1,\frac {1}{4 x^2}\right )\right )}{2 x^2} \]