85.75.3 problem 1 (c)

Internal problem ID [22968]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 329
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 09:17:03 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 34

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

\[ y = c_1 \left (1-x +\frac {1}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-2 x +2 x^{2}-\frac {2}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 35

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

\[ y(x)\to c_1 \left (-x^2+3 x+\frac {1}{x}-3\right )+c_2 \left (\frac {x^2}{3}-x+1\right ) \]

✓ Sympy. Time used: 0.307 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{2700} + \frac {x^{4}}{180} + \frac {x^{3}}{18} + \frac {x^{2}}{3} + x + 1\right ) + O\left (x^{6}\right ) \]