6.11.1 problem 11

Internal problem ID [1840]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:20:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2+x \right ) y^{\prime \prime }+x y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 54

Order:=6; 
ode:=(x+2)*diff(diff(y(x),x),x)+x*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3}{4} x^{2}+\frac {1}{8} x^{3}+\frac {1}{8} x^{4}-\frac {9}{160} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 63

ode=(2+x)*D[y[x],{x,2}]+x*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{40}+\frac {x^4}{12}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {9 x^5}{160}+\frac {x^4}{8}+\frac {x^3}{8}-\frac {3 x^2}{4}+1\right ) \]

✓ Sympy. Time used: 0.347 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{8} + \frac {x^{3}}{8} - \frac {3 x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{3}}{12} - \frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]