12.12.26 problem 31

Internal problem ID [1880]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 31
Date solved : Monday, January 27, 2025 at 05:37:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+2 \alpha y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 55

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

\[ y = \left (1-\alpha \,x^{2}+\frac {\alpha \left (\alpha -2\right ) x^{4}}{6}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha -1\right ) x^{3}}{3}+\frac {\left (\alpha ^{2}-4 \alpha +3\right ) x^{5}}{30}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 78

AsymptoticDSolveValue[D[y[x],{x,2}]-2*x*D[y[x],x]+2*\[Alpha]*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {\alpha ^2 x^5}{30}-\frac {2 \alpha x^5}{15}+\frac {x^5}{10}-\frac {\alpha x^3}{3}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {\alpha ^2 x^4}{6}-\frac {\alpha x^4}{3}-\alpha x^2+1\right ) \]