74.16.9 problem 9

Internal problem ID [16436]
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 : 9
Date solved : Thursday, March 13, 2025 at 08:13:48 AM
CAS classification : [[_2nd_order, _missing_y]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 31

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

\[ y = y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{16} x^{4}-\frac {9}{80} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to c_2 \left (-\frac {9 x^5}{80}+\frac {3 x^4}{16}-\frac {x^3}{4}+x\right )+c_1 \]

✓ Sympy. Time used: 0.724 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {3 x^{3}}{16} - \frac {x^{2}}{4} + 1\right ) + C_{1} + O\left (x^{6}\right ) \]