12.14.14 problem 14

Internal problem ID [1955]
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 : 14
Date solved : Monday, January 27, 2025 at 05:39:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \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: 48

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

\[ y = \frac {c_2 \,x^{{3}/{2}} \left (1-\frac {2}{5} x +\frac {4}{35} x^{2}-\frac {8}{315} x^{3}+\frac {16}{3465} x^{4}-\frac {32}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 86

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {32 x^5}{45045}+\frac {16 x^4}{3465}-\frac {8 x^3}{315}+\frac {4 x^2}{35}-\frac {2 x}{5}+1\right )+\frac {c_2 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )}{x} \]