46.2.17 problem 17

Internal problem ID [9554]
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 : 17
Date solved : Tuesday, September 30, 2025 at 06:20:39 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 52

Order:=8; 
ode:=4*x*diff(diff(y(x),x),x)+1/2*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{7}/{8}} \left (1-\frac {2}{15} x +\frac {2}{345} x^{2}-\frac {4}{32085} x^{3}+\frac {2}{1251315} x^{4}-\frac {4}{294059025} x^{5}+\frac {4}{48519739125} x^{6}-\frac {8}{21397204954125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-2 x +\frac {2}{9} x^{2}-\frac {4}{459} x^{3}+\frac {2}{11475} x^{4}-\frac {4}{1893375} x^{5}+\frac {4}{232885125} x^{6}-\frac {8}{79879597875} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 111

ode=4*x*D[y[x],{x,2}]+1/2*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {8 x^7}{79879597875}+\frac {4 x^6}{232885125}-\frac {4 x^5}{1893375}+\frac {2 x^4}{11475}-\frac {4 x^3}{459}+\frac {2 x^2}{9}-2 x+1\right )+c_1 x^{7/8} \left (-\frac {8 x^7}{21397204954125}+\frac {4 x^6}{48519739125}-\frac {4 x^5}{294059025}+\frac {2 x^4}{1251315}-\frac {4 x^3}{32085}+\frac {2 x^2}{345}-\frac {2 x}{15}+1\right ) \]

✓ Sympy. Time used: 0.329 (sec). Leaf size: 100

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) + y(x) + Derivative(y(x), x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {8 x^{7}}{79879597875} + \frac {4 x^{6}}{232885125} - \frac {4 x^{5}}{1893375} + \frac {2 x^{4}}{11475} - \frac {4 x^{3}}{459} + \frac {2 x^{2}}{9} - 2 x + 1\right ) + C_{1} x^{\frac {7}{8}} \left (\frac {4 x^{6}}{48519739125} - \frac {4 x^{5}}{294059025} + \frac {2 x^{4}}{1251315} - \frac {4 x^{3}}{32085} + \frac {2 x^{2}}{345} - \frac {2 x}{15} + 1\right ) + O\left (x^{8}\right ) \]