12.13.20 problem 23

Internal problem ID [1911]
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 : 23
Date solved : Saturday, March 29, 2025 at 11:43:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

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

\[ y = -1+2 x +\frac {1}{3} x^{3}-\frac {5}{12} x^{4}+\frac {2}{5} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

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

\[ y(x)\to \frac {2 x^5}{5}-\frac {5 x^4}{12}+\frac {x^3}{3}+2 x-1 \]

✓ Sympy. Time used: 0.887 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{5}}{8} + \frac {x^{4}}{8} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {11 x^{4}}{80} - \frac {7 x^{3}}{48} + \frac {x^{2}}{12} + 1\right ) + O\left (x^{6}\right ) \]