12.14.34 problem 36

Internal problem ID [1975]
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 : 36
Date solved : Monday, January 27, 2025 at 05:39:31 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 32

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

\[ y = c_1 \,x^{{1}/{3}} \left (1+\frac {11}{18} x^{2}+\frac {55}{648} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {4}{5} x^{2}+\frac {8}{55} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {55 x^4}{648}+\frac {11 x^2}{18}+1\right )+c_2 \left (\frac {8 x^4}{55}+\frac {4 x^2}{5}+1\right ) \]