40.17.8 problem 18

Internal problem ID [6809]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 18
Date solved : Monday, January 27, 2025 at 02:30:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-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: 37

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

\[ y = c_1 x \left (1-\frac {1}{3} x +\frac {1}{6} x^{2}-\frac {1}{10} x^{3}+\frac {1}{15} x^{4}-\frac {1}{21} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2-2 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 45

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

\[ y(x)\to c_2 \left (\frac {x^5}{15}-\frac {x^4}{10}+\frac {x^3}{6}-\frac {x^2}{3}+x\right )+c_1 \left (\frac {1}{x}+1\right ) \]