Internal problem ID [2307]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page
572
Problem number: Problem 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}+4 x^{2} y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x^{2}\right ) \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,sin(x^2)],y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (x^{2}\right )+c_{2} \cos \left (x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 20
DSolve[y''[x]-1/x*y'[x]+4*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (x^2\right )+c_2 \sin \left (x^2\right ) \\ \end{align*}