73.9.18 problem 14.2 (h)

Internal problem ID [15279]
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 : 14.2 (h)
Date solved : Tuesday, January 28, 2025 at 07:51:30 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-4 x^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 17

dsolve([diff(y(x),x$2)-1/x*diff(y(x),x)-4*x^2*y(x)=0,exp(-x^2)],singsol=all)
 

\[ y = c_{1} \sinh \left (x^{2}\right )+c_{2} \cosh \left (x^{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 23

DSolve[D[y[x],{x,2}]-1/x*D[y[x],x]-4*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \cosh \left (x^2\right )+i c_2 \sinh \left (x^2\right ) \]