52.2.19 problem 19

Internal problem ID [8270]
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 : 19
Date solved : Sunday, March 30, 2025 at 12:50:29 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 52

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

\[ y = c_1 \,x^{{1}/{3}} \left (1+\frac {1}{3} x +\frac {1}{18} x^{2}+\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}+\frac {1}{29160} x^{5}+\frac {1}{524880} x^{6}+\frac {1}{11022480} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1+\frac {1}{2} x +\frac {1}{10} x^{2}+\frac {1}{80} x^{3}+\frac {1}{880} x^{4}+\frac {1}{12320} x^{5}+\frac {1}{209440} x^{6}+\frac {1}{4188800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 113

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {x^7}{11022480}+\frac {x^6}{524880}+\frac {x^5}{29160}+\frac {x^4}{1944}+\frac {x^3}{162}+\frac {x^2}{18}+\frac {x}{3}+1\right )+c_2 \left (\frac {x^7}{4188800}+\frac {x^6}{209440}+\frac {x^5}{12320}+\frac {x^4}{880}+\frac {x^3}{80}+\frac {x^2}{10}+\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 1.048 (sec). Leaf size: 80

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{7}}{4188800} + \frac {x^{6}}{209440} + \frac {x^{5}}{12320} + \frac {x^{4}}{880} + \frac {x^{3}}{80} + \frac {x^{2}}{10} + \frac {x}{2} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {x^{6}}{524880} + \frac {x^{5}}{29160} + \frac {x^{4}}{1944} + \frac {x^{3}}{162} + \frac {x^{2}}{18} + \frac {x}{3} + 1\right ) + O\left (x^{8}\right ) \]