74.16.14 problem 14

Internal problem ID [16449]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 14
Date solved : Monday, March 31, 2025 at 02:53:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

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

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

\[ y = 3-2 x +\frac {5}{3} x^{2}-\frac {2}{9} x^{3}+\frac {1}{18} x^{4}+\frac {1}{135} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 36

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

\[ y(x)\to \frac {x^5}{135}+\frac {x^4}{18}-\frac {2 x^3}{9}+\frac {5 x^2}{3}-2 x+3 \]

✓ Sympy. Time used: 0.770 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{54} + \frac {x^{2}}{3} + 1\right ) + C_{1} x \left (\frac {x^{2}}{9} - \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]