7.14.27 problem 27

Internal problem ID [452]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 27
Date solved : Monday, January 27, 2025 at 02:53:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+\left (2 x^{2}+1\right ) 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: 0.000 (sec). Leaf size: 20

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

\[ y = 1-x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to \frac {x^5}{30}-\frac {x^4}{24}+\frac {x^3}{3}-\frac {x^2}{2}-x+1 \]