12.14.3 problem Example 7.5.3 page 356

Internal problem ID [1944]
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 : Example 7.5.3 page 356
Date solved : Monday, January 27, 2025 at 05:38:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (2 x +2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 47

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

\[ y = c_1 \sqrt {x}\, \left (1+2 x -\frac {9}{8} x^{2}+\frac {7}{4} x^{3}-\frac {607}{640} x^{4}+\frac {13347}{11200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{2} \left (1-\frac {2}{5} x +\frac {27}{35} x^{2}-\frac {34}{105} x^{3}+\frac {584}{1155} x^{4}-\frac {768}{3575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 86

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

\[ y(x)\to c_1 \left (-\frac {768 x^5}{3575}+\frac {584 x^4}{1155}-\frac {34 x^3}{105}+\frac {27 x^2}{35}-\frac {2 x}{5}+1\right ) x^2+c_2 \left (\frac {13347 x^5}{11200}-\frac {607 x^4}{640}+\frac {7 x^3}{4}-\frac {9 x^2}{8}+2 x+1\right ) \sqrt {x} \]