12.16.34 problem 30

Internal problem ID [2096]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 30
Date solved : Monday, January 27, 2025 at 05:42:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \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: 0.008 (sec). Leaf size: 46

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*(1-2*x^2)*diff(y(x),x)-4*(1+2*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1+x^{2}+\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (288 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+144 x^{2}+216 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 45

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*(1-2*x^2)*D[y[x],x]-4*(1+2*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

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