50.21.6 problem 5

Internal problem ID [8143]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.6. Gauss Hypergeometric Equation. Page 187
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:46:35 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.099 (sec). Leaf size: 52

Order:=8; 
ode:=(1-exp(x))*diff(diff(y(x),x),x)+1/2*diff(y(x),x)+exp(x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {1}{4} x +\frac {3}{32} x^{2}+\frac {7}{384} x^{3}+\frac {109}{30720} x^{4}+\frac {13}{24576} x^{5}+\frac {4439}{61931520} x^{6}+\frac {2069}{247726080} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-2 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 109

ode=(1-Exp[x])*D[y[x],{x,2}]+1/2*D[y[x],x]+Exp[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {x^7}{2520}-\frac {x^6}{360}-\frac {x^5}{60}-\frac {x^4}{12}-\frac {x^3}{3}-x^2-2 x+1\right )+c_1 \left (\frac {2069 x^7}{247726080}+\frac {4439 x^6}{61931520}+\frac {13 x^5}{24576}+\frac {109 x^4}{30720}+\frac {7 x^3}{384}+\frac {3 x^2}{32}+\frac {x}{4}+1\right ) x^{3/2} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - exp(x))*Derivative(y(x), (x, 2)) + y(x)*exp(x) + Derivative(y(x), x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE (1 - exp(x))*Derivative(y(x), (x, 2)) + y(x)*exp(x) + Derivative(y(x), x)/2 does not match hint 2nd_power_series_regular