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 : Tuesday, March 04, 2025 at 01:46:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 48

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

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

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

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

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

✓ Sympy. Time used: 0.919 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + x*(2*x + 3)*Derivative(y(x), x) - (1 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {16 x^{4}}{3465} - \frac {8 x^{3}}{315} + \frac {4 x^{2}}{35} - \frac {2 x}{5} + 1\right ) + \frac {C_{1} \left (\frac {x^{6}}{720} - \frac {x^{5}}{120} + \frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right )}{x} + O\left (x^{6}\right ) \]