12.15.43 problem 39

Internal problem ID [2041]
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 II. Exercises 7.6. Page 374
Problem number : 39
Date solved : Monday, January 27, 2025 at 05:40:55 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }+\left (12 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: 36

Order:=6; 
dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(3+8*x^2)*diff(y(x),x)+(1+12*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 71

AsymptoticDSolveValue[x^2*(1+x^2)*D[y[x],{x,2}]+x*(3+8*x^2)*D[y[x],x]+(1+12*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (\frac {15 x^4}{8}-\frac {3 x^2}{2}+1\right )}{x}+c_2 \left (\frac {\frac {x^2}{4}-\frac {13 x^4}{32}}{x}+\frac {\left (\frac {15 x^4}{8}-\frac {3 x^2}{2}+1\right ) \log (x)}{x}\right ) \]