7.16.7 problem 7

Internal problem ID [504]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 7
Date solved : Saturday, March 29, 2025 at 04:55:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 45

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

\[ y = c_1 x \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (\frac {27 x^2}{8}+\frac {1}{x^2}-\frac {9 x}{2}-\frac {3}{x}+\frac {9}{2}\right )+c_2 \left (\frac {27 x^5}{280}-\frac {9 x^4}{40}+\frac {9 x^3}{20}-\frac {3 x^2}{4}+x\right ) \]

✓ Sympy. Time used: 0.885 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {27 x^{4}}{280} - \frac {9 x^{3}}{40} + \frac {9 x^{2}}{20} - \frac {3 x}{4} + 1\right ) + \frac {C_{1} \left (\frac {9 x^{2}}{2} - 3 x + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]