Internal problem ID [6176]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 52
Order:=8; dsolve(2*x*diff(y(x),x$2)+(1+2*x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {5}{3} x +\frac {7}{6} x^{2}-\frac {1}{2} x^{3}+\frac {11}{72} x^{4}-\frac {13}{360} x^{5}+\frac {1}{144} x^{6}-\frac {17}{15120} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1-4 x +4 x^{2}-\frac {32}{15} x^{3}+\frac {16}{21} x^{4}-\frac {64}{315} x^{5}+\frac {64}{1485} x^{6}-\frac {1024}{135135} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 109
AsymptoticDSolveValue[2*x*y''[x]+(1+2*x)*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {17 x^7}{15120}+\frac {x^6}{144}-\frac {13 x^5}{360}+\frac {11 x^4}{72}-\frac {x^3}{2}+\frac {7 x^2}{6}-\frac {5 x}{3}+1\right )+c_2 \left (-\frac {1024 x^7}{135135}+\frac {64 x^6}{1485}-\frac {64 x^5}{315}+\frac {16 x^4}{21}-\frac {32 x^3}{15}+4 x^2-4 x+1\right ) \]