36.5.1 problem 1

Internal problem ID [6345]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 1
Date solved : Monday, January 27, 2025 at 01:57:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+x \right ) y^{\prime \prime }-x^{2} y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 54

Order:=6; 
dsolve((x+1)*diff(y(x),x$2)-x^2*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{8} x^{4}-\frac {3}{10} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {1}{3} x^{4}-\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 63

AsymptoticDSolveValue[(x+1)*D[y[x],{x,2}]-x^2*D[y[x],x]+3*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (-\frac {x^5}{8}+\frac {x^4}{3}-\frac {x^3}{2}+x\right )+c_1 \left (-\frac {3 x^5}{10}+\frac {x^4}{8}+\frac {x^3}{2}-\frac {3 x^2}{2}+1\right ) \]