Internal problem ID [6209]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+8 x \left (1+x \right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 81
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+8*x*(x+1)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+x -\frac {1}{4} x^{2}+\frac {1}{12} x^{3}-\frac {5}{192} x^{4}+\frac {7}{960} x^{5}-\frac {7}{3840} x^{6}+\frac {11}{26880} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\left (-4\right ) x +\frac {3}{4} x^{2}-\frac {1}{4} x^{3}+\frac {31}{384} x^{4}-\frac {3}{128} x^{5}+\frac {419}{69120} x^{6}-\frac {97}{69120} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 166
AsymptoticDSolveValue[4*x^2*y''[x]+8*x*(x+1)*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (\frac {11 x^7}{26880}-\frac {7 x^6}{3840}+\frac {7 x^5}{960}-\frac {5 x^4}{192}+\frac {x^3}{12}-\frac {x^2}{4}+x+1\right )}{\sqrt {x}}+c_2 \left (\frac {-\frac {97 x^7}{69120}+\frac {419 x^6}{69120}-\frac {3 x^5}{128}+\frac {31 x^4}{384}-\frac {x^3}{4}+\frac {3 x^2}{4}-4 x}{\sqrt {x}}+\frac {\left (\frac {11 x^7}{26880}-\frac {7 x^6}{3840}+\frac {7 x^5}{960}-\frac {5 x^4}{192}+\frac {x^3}{12}-\frac {x^2}{4}+x+1\right ) \log (x)}{\sqrt {x}}\right ) \]