12.14.36 problem 38

Internal problem ID [1977]
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 : 38
Date solved : Monday, January 27, 2025 at 05:39:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 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: 35

Order:=6; 
dsolve(3*x^2*(2-x^2)*diff(y(x),x$2)+x*(1-11*x^2)*diff(y(x),x)+(1-5*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1+\frac {4}{11} x^{2}+\frac {40}{253} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1+\frac {3}{8} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 52

AsymptoticDSolveValue[3*x^2*(2-x^2)*D[y[x],{x,2}]+x*(1-11*x^2)*D[y[x],x]+(1-5*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {21 x^4}{128}+\frac {3 x^2}{8}+1\right )+c_2 \sqrt [3]{x} \left (\frac {40 x^4}{253}+\frac {4 x^2}{11}+1\right ) \]