12.16.41 problem 37

Internal problem ID [2103]
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 : 37
Date solved : Monday, January 27, 2025 at 05:42:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(5-x^2)*diff(y(x),x)-(7+25*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{7} \left (1-\frac {6}{5} x^{2}+\frac {7}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-203212800+406425600 x^{2}-609638400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 40

AsymptoticDSolveValue[x^2*(1+x^2)*D[y[x],{x,2}]-x*(5-x^2)*D[y[x],x]-(7+25*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (3 x^3-2 x+\frac {1}{x}\right )+c_2 \left (\frac {7 x^{11}}{5}-\frac {6 x^9}{5}+x^7\right ) \]