28.5.15 problem 9.15

Internal problem ID [4602]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.15
Date solved : Monday, January 27, 2025 at 09:26:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 43

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

\[ y \left (x \right ) = c_{1} x \left (1+\frac {2}{3} x +\frac {1}{4} x^{2}+\frac {1}{15} x^{3}+\frac {1}{72} x^{4}+\frac {1}{420} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2+x^{2}+\frac {2}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^3}{8}-\frac {x^2}{3}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^5}{72}+\frac {x^4}{15}+\frac {x^3}{4}+\frac {2 x^2}{3}+x\right ) \]