39.6.10 problem Problem 27.48

Internal problem ID [6568]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 27. Power series solutions of linear DE with variable coefficients. Supplementary Problems. page 274
Problem number : Problem 27.48
Date solved : Monday, January 27, 2025 at 02:12:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 18

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

\[ y = 1-x -\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 29

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

\[ y(x)\to -\frac {x^5}{20}-\frac {x^4}{12}-\frac {x^3}{3}-x+1 \]