85.72.5 problem 1 (e)

Internal problem ID [22941]
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 (e)
Date solved : Thursday, October 02, 2025 at 09:16:48 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x y^{\prime \prime }+y^{\prime }&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+diff(y(x),x) = 0; 
ic:=[y(1) = 2, D(y)(1) = 3]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

\[ y(x)\to \frac {3}{5} (x-1)^5-\frac {3}{4} (x-1)^4+(x-1)^3-\frac {3}{2} (x-1)^2+3 (x-1)+2 \]

✓ Sympy. Time used: 0.152 (sec). Leaf size: 12

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

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