Internal problem ID [6223]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a
Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (3+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: 0.11 (sec). Leaf size: 50
Order:=8; dsolve(x*diff(y(x),x$2)+(3+2*x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1-\frac {4}{3} x +x^{2}-\frac {8}{15} x^{3}+\frac {2}{9} x^{4}-\frac {8}{105} x^{5}+\frac {1}{45} x^{6}-\frac {16}{2835} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2+4 x^{2}-\frac {16}{3} x^{3}+4 x^{4}-\frac {32}{15} x^{5}+\frac {8}{9} x^{6}-\frac {32}{105} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 77
AsymptoticDSolveValue[x*y''[x]+(3+2*x)*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {4 x^4}{9}+\frac {16 x^3}{15}-2 x^2+\frac {1}{x^2}+\frac {8 x}{3}-2\right )+c_2 \left (\frac {x^6}{45}-\frac {8 x^5}{105}+\frac {2 x^4}{9}-\frac {8 x^3}{15}+x^2-\frac {4 x}{3}+1\right ) \]