52.4.7 problem 15

Internal problem ID [8317]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW. Page 271
Problem number : 15
Date solved : Wednesday, March 05, 2025 at 05:34:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x 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: 0.001 (sec). Leaf size: 24

Order:=8; 
ode:=diff(diff(y(x),x),x)+x*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);
 

\[ y = 3-2 x -3 x^{2}+x^{3}+x^{4}-\frac {1}{4} x^{5}-\frac {1}{5} x^{6}+\frac {1}{24} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode=D[y[x],{x,2}]+x*D[y[x],x]+2*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==-2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {x^7}{24}-\frac {x^6}{5}-\frac {x^5}{4}+x^4+x^3-3 x^2-2 x+3 \]

✓ Sympy. Time used: 0.785 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + 2*y(x) + Derivative(y(x), (x, 2)),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=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{15} + \frac {x^{4}}{3} - x^{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{8} - \frac {x^{2}}{2} + 1\right ) + O\left (x^{8}\right ) \]