36.6.22 problem 25

Internal problem ID [6383]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 25
Date solved : Monday, January 27, 2025 at 01:58:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y&=\cos \left (x \right ) \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 38

Order:=6; 
dsolve((1+x^2)*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=cos(x),y(x),type='series',x=0);
 

\[ y = \left (\frac {1}{24} x^{4}-\frac {1}{2} x^{2}+1\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +\frac {x^{2}}{2}-\frac {x^{4}}{12}+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 41

AsymptoticDSolveValue[(1+x^2)*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==Cos[x],y[x],{x,0,"6"-1}]
 

\[ y(x)\to -\frac {x^4}{12}+\frac {x^2}{2}+c_1 \left (\frac {x^4}{24}-\frac {x^2}{2}+1\right )+c_2 x \]