7.14.10 problem 10

Internal problem ID [435]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 11:24:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 y^{\prime \prime }+x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 39

Order:=6; 
ode:=3*diff(diff(y(x),x),x)+x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {2}{3} x^{2}+\frac {1}{27} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{360} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode=D[y[x],{x,2}]+x*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{40}+\frac {x^3}{2}+x\right )+c_1 \left (\frac {x^4}{3}+2 x^2+1\right ) \]

✓ Sympy. Time used: 0.741 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - 4*y(x) + 3*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{27} + \frac {2 x^{2}}{3} + 1\right ) + C_{1} x \left (\frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]