52.1.22 problem 20

Internal problem ID [8239]
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. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 20
Date solved : Sunday, March 30, 2025 at 12:49:25 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 24

Order:=8; 
ode:=(1+x)*diff(diff(y(x),x),x)-(2-x)*diff(y(x),x)+y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = -1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 2-x -2 x^{2}-\frac {1}{3} x^{3}+\frac {1}{2} x^{4}-\frac {1}{30} x^{5}-\frac {13}{180} x^{6}+\frac {1}{28} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 48

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

\[ y(x)\to \frac {x^7}{28}-\frac {13 x^6}{180}-\frac {x^5}{30}+\frac {x^4}{2}-\frac {x^3}{3}-2 x^2-x+2 \]

✓ Sympy. Time used: 0.981 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {17 x^{6}}{720} + \frac {x^{5}}{120} + \frac {x^{4}}{8} - \frac {x^{3}}{6} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{5}}{40} + \frac {x^{4}}{20} - \frac {x^{3}}{4} + x + 1\right ) + O\left (x^{8}\right ) \]