20.11.5 problem Problem 5

Internal problem ID [3787]
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
Date solved : Monday, January 27, 2025 at 08:02:18 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&=\sin \left (x^{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.001 (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)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 20

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

\[ y(x)\to c_1 \cos \left (x^2\right )+c_2 \sin \left (x^2\right ) \]