73.24.10 problem 34.5 (j)

Internal problem ID [15690]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.5 (j)
Date solved : Tuesday, January 28, 2025 at 08:05:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (\pi \,x^{2}\right ) y^{\prime \prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 51

Order:=6; 
dsolve(sin(Pi*x^2)*diff(y(x),x$2)+x^2*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{2}}{2 \pi }+\frac {x^{4}}{24 \pi ^{2}}\right ) y \left (0\right )+\left (x -\frac {x^{3}}{6 \pi }+\frac {x^{5}}{120 \pi ^{2}}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 54

AsymptoticDSolveValue[Sin[Pi*x^2]*D[y[x],{x,2}]+x^2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{120 \pi ^2}-\frac {x^3}{6 \pi }+x\right )+c_1 \left (\frac {x^4}{24 \pi ^2}-\frac {x^2}{2 \pi }+1\right ) \]