3.16 problem 16

Internal problem ID [6151]

Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section: CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number: 16.
ODE order: 1.
ODE degree: 3.

CAS Maple gives this as type [[_3rd_order, _exact, _linear, _homogeneous]]

Solution by Maple

Time used: 0.006 (sec). Leaf size: 44

Order:=8; 
dsolve(diff(y(x),x$3)+x^2*diff(y(x),x$2)+5*x*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = \left (1-\frac {1}{2} x^{3}+\frac {1}{10} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{3} x^{4}+\frac {1}{18} x^{7}\right ) D\relax (y )\relax (0)+\frac {D^{\relax (2)}\relax (y )\relax (0) x^{2}}{2}-\frac {D^{\relax (2)}\relax (y )\relax (0) x^{5}}{8}+O\left (x^{8}\right ) \]

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 60

AsymptoticDSolveValue[y'''[x]+x^2*y''[x]+5*x*y'[x]+3*y[x]==0,y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {x^7}{18}-\frac {x^4}{3}+x\right )+c_1 \left (\frac {x^6}{10}-\frac {x^3}{2}+1\right )+c_3 \left (\frac {x^2}{2}-\frac {x^5}{8}\right ) \]