76.12.21 problem 33

Internal problem ID [17577]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 33
Date solved : Tuesday, January 28, 2025 at 10:44:26 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{3} 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: 0.005 (sec). Leaf size: 17

dsolve([x*diff(y(x),x$2)-diff(y(x),x)+4*x^3*y(x)=0,sin(x^2)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 20

DSolve[x*D[y[x],{x,2}]-D[y[x],x]+4*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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