52.2.17 problem 17

Internal problem ID [8268]
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 : Monday, January 27, 2025 at 03:47:29 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*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 111

AsymptoticDSolveValue[4*x*D[y[x],{x,2}]+1/2*D[y[x],x]+y[x]==0,y[x],{x,0,"8"-1}]
 

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