46.2.15 problem 15

Internal problem ID [9552]
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 : 15
Date solved : Tuesday, September 30, 2025 at 06:20:37 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x y^{\prime \prime }-y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 52

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

\[ y = c_1 \,x^{{3}/{2}} \left (1-\frac {2}{5} x +\frac {2}{35} x^{2}-\frac {4}{945} x^{3}+\frac {2}{10395} x^{4}-\frac {4}{675675} x^{5}+\frac {4}{30405375} x^{6}-\frac {8}{3618239625} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1+2 x -2 x^{2}+\frac {4}{9} x^{3}-\frac {2}{45} x^{4}+\frac {4}{1575} x^{5}-\frac {4}{42525} x^{6}+\frac {8}{3274425} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 109

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

\[ y(x)\to c_2 \left (\frac {8 x^7}{3274425}-\frac {4 x^6}{42525}+\frac {4 x^5}{1575}-\frac {2 x^4}{45}+\frac {4 x^3}{9}-2 x^2+2 x+1\right )+c_1 \left (-\frac {8 x^7}{3618239625}+\frac {4 x^6}{30405375}-\frac {4 x^5}{675675}+\frac {2 x^4}{10395}-\frac {4 x^3}{945}+\frac {2 x^2}{35}-\frac {2 x}{5}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.315 (sec). Leaf size: 92

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

\[ y{\left (x \right )} = C_{2} \left (\frac {8 x^{7}}{3274425} - \frac {4 x^{6}}{42525} + \frac {4 x^{5}}{1575} - \frac {2 x^{4}}{45} + \frac {4 x^{3}}{9} - 2 x^{2} + 2 x + 1\right ) + C_{1} x^{\frac {3}{2}} \left (- \frac {4 x^{5}}{675675} + \frac {2 x^{4}}{10395} - \frac {4 x^{3}}{945} + \frac {2 x^{2}}{35} - \frac {2 x}{5} + 1\right ) + O\left (x^{8}\right ) \]