6.13.25 problem 28

Internal problem ID [1916]
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 : 28
Date solved : Tuesday, September 30, 2025 at 05:21:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (6+4 x \right ) y^{\prime \prime }+\left (1+2 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = -1+2 \left (x +1\right )-\frac {1}{4} \left (x +1\right )^{2}+\frac {1}{2} \left (x +1\right )^{3}-\frac {65}{96} \left (x +1\right )^{4}+\frac {67}{80} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 46

ode=(6+4*x)*D[y[x],{x,2}]+(1+2*x)*y[x]==0; 
ic={y[-1]==-1,Derivative[1][y][-1]==2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to \frac {67}{80} (x+1)^5-\frac {65}{96} (x+1)^4+\frac {1}{2} (x+1)^3-\frac {1}{4} (x+1)^2+2 (x+1)-1 \]

✓ Sympy. Time used: 0.255 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = C_{2} \left (\frac {11 \left (x + 1\right )^{4}}{32} - \frac {\left (x + 1\right )^{3}}{3} + \frac {\left (x + 1\right )^{2}}{4} + 1\right ) + C_{1} \left (x - \frac {\left (x + 1\right )^{4}}{6} + \frac {\left (x + 1\right )^{3}}{12} + 1\right ) + O\left (x^{6}\right ) \]