Internal problem ID [13223]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}-4 y x^{2}=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x^{2}} \end {align*}
✓ Solution by Maple
Time used: 0.0 (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)],y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sinh \left (x^{2}\right )+c_{2} \cosh \left (x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 23
DSolve[y''[x]-1/x*y'[x]-4*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cosh \left (x^2\right )+i c_2 \sinh \left (x^2\right ) \]