85.76.1 problem 2

Internal problem ID [22979]
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 : 2
Date solved : Thursday, October 02, 2025 at 09:17:11 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-x y^{\prime }-y&=5 \sqrt {x} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 42

Order:=6; 
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x) = 5*x^(1/2); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1+\frac {1}{3} x^{2}+\frac {1}{15} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{{5}/{2}} \left (\frac {4}{3}+\frac {8}{27} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 101

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

\[ y(x)\to c_2 \left (\frac {x^5}{15}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {x^4}{8}+\frac {x^2}{2}+1\right )+\frac {10}{3} \left (\frac {x^5}{15}+\frac {x^3}{3}+x\right ) x^{3/2}+\left (\frac {x^4}{8}+\frac {x^2}{2}+1\right ) \left (\frac {5 x^{9/2}}{27}-2 x^{5/2}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*sqrt(x) - x*Derivative(y(x), x) - y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -5*sqrt(x) - x*Derivative(y(x), x) - y(x) + Derivative(y(x), (x, 2)) does not match