12.13.41 problem 40

Internal problem ID [1932]
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 : 40
Date solved : Tuesday, March 04, 2025 at 01:46:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1+2 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 )&=3 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

\[ y = -2+3 x +x^{2}-\frac {1}{6} x^{3}-\frac {3}{4} x^{4}+\frac {31}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

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

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

\[ y(x)\to \frac {31 x^5}{120}-\frac {3 x^4}{4}-\frac {x^3}{6}+x^2+3 x-2 \]

✓ Sympy. Time used: 0.886 (sec). Leaf size: 39

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

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