85.76.2 problem 3

Internal problem ID [22980]
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. B Exercises at page 330
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:17:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+y x&=2 x \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 44

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

\[ y = c_1 \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x}+x^{2} \left (\frac {1}{3}-\frac {1}{60} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 143

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

\[ y(x)\to \left (-\frac {x^6}{5040}+\frac {x^4}{120}-\frac {x^2}{6}+1\right ) \left (\frac {x^6}{72}-\frac {x^4}{4}+x^2\right )+\frac {c_2 \left (-\frac {x^6}{720}+\frac {x^4}{24}-\frac {x^2}{2}+1\right )}{x}+c_1 \left (-\frac {x^6}{5040}+\frac {x^4}{120}-\frac {x^2}{6}+1\right )+\frac {\left (\frac {x^5}{15}-\frac {2 x^3}{3}\right ) \left (-\frac {x^6}{720}+\frac {x^4}{24}-\frac {x^2}{2}+1\right )}{x} \]

✗ Sympy

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

ValueError : ODE x*y(x) + x*Derivative(y(x), (x, 2)) - 2*x + 2*Derivative(y(x), x) does not match hi