12.16.44 problem 40

Internal problem ID [2106]
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 : 40
Date solved : Monday, January 27, 2025 at 05:42:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 36

Order:=6; 
dsolve(4*x^2*(1+x^2)*diff(y(x),x$2)+4*x*(2+x^2)*diff(y(x),x)-(15+x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {1}{6} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-144-216 x^{2}-54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{5}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 58

AsymptoticDSolveValue[4*x^2*(1+x^2)*D[y[x],{x,2}]+4*x*(2+x^2)*D[y[x],x]-(15+x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {3 x^{3/2}}{8}+\frac {1}{x^{5/2}}+\frac {3}{2 \sqrt {x}}\right )+c_2 \left (\frac {x^{11/2}}{16}-\frac {x^{7/2}}{6}+x^{3/2}\right ) \]