85.72.10 problem 1 (j)

Internal problem ID [22946]
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 316
Problem number : 1 (j)
Date solved : Thursday, October 02, 2025 at 09:16:50 PM
CAS classification : [[_2nd_order, _missing_y]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 45

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

\[ y = y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{4}-\frac {\left (x -1\right )^{4}}{8}+\frac {\left (x -1\right )^{5}}{16}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 48

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

\[ y(x)\to c_2 \left (\frac {1}{16} (x-1)^5-\frac {1}{8} (x-1)^4+\frac {1}{4} (x-1)^3-\frac {1}{2} (x-1)^2+x-1\right )+c_1 \]

✓ Sympy. Time used: 0.220 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {C_{2} \left (- 4 x - \left (x - 1\right )^{3} + 2 \left (x - 1\right )^{2} + 12\right )}{8} + C_{1} \left (x - 1\right ) + O\left (x^{6}\right ) \]