Internal problem ID [13527]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.1 (b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }-4 y=0} \]
✓ Solution by Maple
Time used: 0.359 (sec). Leaf size: 54
dsolve(diff(y(x),x$2)+x^2*diff(y(x),x)-4*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{3}}{3}} \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 35
DSolve[y''[x]+x^2*y'[x]-4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 e^{-\frac {x^3}{3}} \text {HeunT}[4,-2,0,0,-1,x]+c_1 \text {HeunT}[4,0,0,0,1,x] \]