12.13.16 problem 19

Internal problem ID [1907]
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 : 19
Date solved : Tuesday, March 04, 2025 at 01:46:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-7 \end{align*}

✓ Maple. Time used: 0.000 (sec). Leaf size: 20

Order:=6; 
ode:=(2+4*x)*diff(diff(y(x),x),x)-4*diff(y(x),x)-(6+4*x)*y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = -7; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 2-7 x -4 x^{2}-\frac {17}{6} x^{3}-\frac {3}{4} x^{4}-\frac {9}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

ode=(2+4*x)*D[y[x],{x,2}]-4*D[y[x],x]-(6+4*x)*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==-7}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {9 x^5}{40}-\frac {3 x^4}{4}-\frac {17 x^3}{6}-4 x^2-7 x+2 \]

✓ Sympy. Time used: 0.950 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*x + 2)*Derivative(y(x), (x, 2)) - (4*x + 6)*y(x) - 4*Derivative(y(x), x),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): -7} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {5 x^{4}}{24} + \frac {x^{3}}{3} + \frac {3 x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + O\left (x^{6}\right ) \]