7.15.21 problem 21

Internal problem ID [477]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 21
Date solved : Monday, January 27, 2025 at 02:54:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} 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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 33

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

\[ y = \frac {c_1 \left (1+x^{2}+\frac {1}{10} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1+\frac {1}{7} x^{2}+\frac {1}{154} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 44

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

\[ y(x)\to c_1 x \left (\frac {x^4}{154}+\frac {x^2}{7}+1\right )+\frac {c_2 \left (\frac {x^4}{10}+x^2+1\right )}{\sqrt {x}} \]