12.13.21 problem 24

Internal problem ID [1912]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 01:46:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (-1\right )&=2\\ y^{\prime }\left (-1\right )&=-3 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
ode:=(2*x+3)*diff(diff(y(x),x),x)+3*diff(y(x),x)-x*y(x) = 0; 
ic:=y(-1) = 2, D(y)(-1) = -3; 
dsolve([ode,ic],y(x),type='series',x=-1);
 

\[ y = 2-3 \left (x +1\right )+\frac {7}{2} \left (x +1\right )^{2}-5 \left (x +1\right )^{3}+\frac {197}{24} \left (x +1\right )^{4}-\frac {287}{20} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 44

ode=(3+2*x)*D[y[x],{x,2}]+3*D[y[x],x]-x*y[x]==0; 
ic={y[-1]==2,Derivative[1][y][-1]==-3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to -\frac {287}{20} (x+1)^5+\frac {197}{24} (x+1)^4-5 (x+1)^3+\frac {7}{2} (x+1)^2-3 (x+1)+2 \]

✓ Sympy. Time used: 0.821 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + (2*x + 3)*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), x),0) 
ics = {y(-1): 2, Subs(Derivative(y(x), x), x, -1): -3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {31 \left (x + 1\right )^{4}}{8} + \frac {7 \left (x + 1\right )^{3}}{3} - \frac {3 \left (x + 1\right )^{2}}{2} + 1\right ) + C_{1} \left (- \frac {41 \left (x + 1\right )^{4}}{24} + \left (x + 1\right )^{3} - \frac {\left (x + 1\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]