12.11.6 problem 16

Internal problem ID [1845]
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 : 16
Date solved : Tuesday, March 04, 2025 at 01:44:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (4+2 x \right ) y^{\prime }+\left (2+x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -1 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 74

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+(4+2*x)*diff(y(x),x)+(x+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=-1);
 

\[ y = \left (1+\frac {\left (x +1\right )^{2}}{2}+\frac {2 \left (x +1\right )^{3}}{3}+\frac {7 \left (x +1\right )^{4}}{8}+\frac {17 \left (x +1\right )^{5}}{15}\right ) y \left (-1\right )+\left (x +1+\left (x +1\right )^{2}+\frac {3 \left (x +1\right )^{3}}{2}+2 \left (x +1\right )^{4}+\frac {103 \left (x +1\right )^{5}}{40}\right ) y^{\prime }\left (-1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 81

ode=(x)*D[y[x],{x,2}]+(4+2*x)*D[y[x],x]+(2+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to c_1 \left (\frac {17}{15} (x+1)^5+\frac {7}{8} (x+1)^4+\frac {2}{3} (x+1)^3+\frac {1}{2} (x+1)^2+1\right )+c_2 \left (\frac {103}{40} (x+1)^5+2 (x+1)^4+\frac {3}{2} (x+1)^3+(x+1)^2+x+1\right ) \]

✓ Sympy. Time used: 0.787 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (x + 2)*y(x) + (2*x + 4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x + 2 \left (x + 1\right )^{4} + \frac {3 \left (x + 1\right )^{3}}{2} + \left (x + 1\right )^{2} + 1\right ) + C_{1} \left (\frac {7 \left (x + 1\right )^{4}}{8} + \frac {2 \left (x + 1\right )^{3}}{3} + \frac {\left (x + 1\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]