52.4.10 problem 18

Internal problem ID [8320]
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 : 18
Date solved : Wednesday, March 05, 2025 at 05:35:03 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-6\\ y^{\prime }\left (1\right )&=3 \end{align*}

✓ Maple. Time used: 0.000 (sec). Leaf size: 22

Order:=8; 
ode:=diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0; 
ic:=y(1) = -6, D(y)(1) = 3; 
dsolve([ode,ic],y(x),type='series',x=1);
 

\[ y = -6+3 \left (x -1\right )+\frac {3}{2} \left (x -1\right )^{2}-\frac {3}{2} \left (x -1\right )^{3}+\frac {3}{10} \left (x -1\right )^{5}-\frac {1}{20} \left (x -1\right )^{6}-\frac {1}{28} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 55

ode=D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0; 
ic={y[1]==-6,Derivative[1][y][1]==3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,7}]
 

\[ y(x)\to -\frac {1}{28} (x-1)^7-\frac {1}{20} (x-1)^6+\frac {3}{10} (x-1)^5-\frac {3}{2} (x-1)^3+\frac {3}{2} (x-1)^2+3 (x-1)-6 \]

✓ Sympy. Time used: 0.826 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(1): -6, Subs(Derivative(y(x), x), x, 1): 3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {\left (x - 1\right )^{6}}{180} - \frac {\left (x - 1\right )^{5}}{20} + \frac {\left (x - 1\right )^{4}}{12} + \frac {\left (x - 1\right )^{3}}{6} - \frac {\left (x - 1\right )^{2}}{2} + 1\right ) + C_{1} \left (x - \frac {\left (x - 1\right )^{6}}{36} + \frac {\left (x - 1\right )^{4}}{6} - \frac {\left (x - 1\right )^{3}}{6} - \frac {\left (x - 1\right )^{2}}{2} - 1\right ) + O\left (x^{8}\right ) \]