12.15.38 problem 34

Internal problem ID [2036]
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 : 34
Date solved : Monday, January 27, 2025 at 05:40:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 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: 36

Order:=6; 
dsolve(4*x^2*(4+x^2)*diff(y(x),x$2)+3*x*(8+3*x^2)*diff(y(x),x)+(1-9*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {5}{32} x^{2}-\frac {15}{2048} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {13}{64} x^{2}+\frac {13}{8192} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{{1}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 77

AsymptoticDSolveValue[4*x^2*(4+x^2)*D[y[x],{x,2}]+3*x*(8+3*x^2)*D[y[x],x]+(1-9*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {15 x^4}{2048}+\frac {5 x^2}{32}+1\right )}{\sqrt [4]{x}}+c_2 \left (\frac {\frac {13 x^4}{8192}-\frac {13 x^2}{64}}{\sqrt [4]{x}}+\frac {\left (-\frac {15 x^4}{2048}+\frac {5 x^2}{32}+1\right ) \log (x)}{\sqrt [4]{x}}\right ) \]