Internal problem ID [5845]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }-y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 52
Order:=8; dsolve(2*x*diff(y(x),x$2)-diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {3}{2}} \left (1-\frac {2}{5} x +\frac {2}{35} x^{2}-\frac {4}{945} x^{3}+\frac {2}{10395} x^{4}-\frac {4}{675675} x^{5}+\frac {4}{30405375} x^{6}-\frac {8}{3618239625} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1+2 x -2 x^{2}+\frac {4}{9} x^{3}-\frac {2}{45} x^{4}+\frac {4}{1575} x^{5}-\frac {4}{42525} x^{6}+\frac {8}{3274425} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 109
AsymptoticDSolveValue[2*x*y''[x]-y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {8 x^7}{3274425}-\frac {4 x^6}{42525}+\frac {4 x^5}{1575}-\frac {2 x^4}{45}+\frac {4 x^3}{9}-2 x^2+2 x+1\right )+c_1 \left (-\frac {8 x^7}{3618239625}+\frac {4 x^6}{30405375}-\frac {4 x^5}{675675}+\frac {2 x^4}{10395}-\frac {4 x^3}{945}+\frac {2 x^2}{35}-\frac {2 x}{5}+1\right ) x^{3/2} \]