7.15.25 problem 25

Internal problem ID [481]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 25
Date solved : Saturday, March 29, 2025 at 04:54:40 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{2} x +\frac {1}{8} x^{2}-\frac {1}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-x +\frac {1}{3} x^{2}-\frac {1}{15} x^{3}+\frac {1}{105} x^{4}-\frac {1}{945} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{3840}+\frac {x^4}{384}-\frac {x^3}{48}+\frac {x^2}{8}-\frac {x}{2}+1\right )+c_2 \left (-\frac {x^5}{945}+\frac {x^4}{105}-\frac {x^3}{15}+\frac {x^2}{3}-x+1\right ) \]

✓ Sympy. Time used: 0.918 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{5}}{945} + \frac {x^{4}}{105} - \frac {x^{3}}{15} + \frac {x^{2}}{3} - x + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{384} - \frac {x^{3}}{48} + \frac {x^{2}}{8} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]