52.2.7 problem 7

Internal problem ID [8258]
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. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 7
Date solved : Sunday, March 30, 2025 at 12:50:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+x -6\right ) y^{\prime \prime }+\left (x +3\right ) y^{\prime }+\left (x -2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 79

Order:=8; 
ode:=(x^2+x-6)*diff(diff(y(x),x),x)+(x+3)*diff(y(x),x)+(x-2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}-\frac {139}{116640} x^{6}-\frac {5377}{9797760} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}+\frac {619}{155520} x^{6}+\frac {689}{408240} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 98

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

\[ y(x)\to c_1 \left (-\frac {5377 x^7}{9797760}-\frac {139 x^6}{116640}-\frac {7 x^5}{2160}-\frac {17 x^4}{2592}-\frac {x^3}{108}-\frac {x^2}{6}+1\right )+c_2 \left (\frac {689 x^7}{408240}+\frac {619 x^6}{155520}+\frac {13 x^5}{1440}+\frac {23 x^4}{864}+\frac {x^3}{36}+\frac {x^2}{4}+x\right ) \]

✓ Sympy. Time used: 1.051 (sec). Leaf size: 73

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {139 x^{6}}{116640} - \frac {7 x^{5}}{2160} - \frac {17 x^{4}}{2592} - \frac {x^{3}}{108} - \frac {x^{2}}{6} + 1\right ) + C_{1} x \left (\frac {619 x^{5}}{155520} + \frac {13 x^{4}}{1440} + \frac {23 x^{3}}{864} + \frac {x^{2}}{36} + \frac {x}{4} + 1\right ) + O\left (x^{8}\right ) \]