12.14.40 problem 42

Internal problem ID [1981]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 42
Date solved : Monday, January 27, 2025 at 05:39:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+5 x \left (x^{2}+1\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

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

Order:=6; 
dsolve(3*x^2*(1+x^2)*diff(y(x),x$2)+5*x*(1+x^2)*diff(y(x),x)-(1-5*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1-\frac {3}{10} x^{2}+\frac {39}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {3}{2} x^{2}+\frac {15}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 50

AsymptoticDSolveValue[3*x^2*(1+x^2)*D[y[x],{x,2}]+5*x*(1+x^2)*D[y[x],x]-(1-5*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {39 x^4}{320}-\frac {3 x^2}{10}+1\right )+\frac {c_2 \left (\frac {15 x^4}{32}-\frac {3 x^2}{2}+1\right )}{x} \]