Internal problem ID [6271]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 23.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (x +3\right ) y^{\prime }+\left (2 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 81
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(3+x)*diff(y(x),x)+(1+2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (x -\frac {3}{4} x^{2}+\frac {11}{36} x^{3}-\frac {25}{288} x^{4}+\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}+\frac {121}{235200} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}-\frac {1}{5040} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 162
AsymptoticDSolveValue[x^2*y''[x]+x*(3+x)*y'[x]+(1+2*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {x^7}{5040}+\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )}{x}+c_2 \left (\frac {\frac {121 x^7}{235200}-\frac {49 x^6}{14400}+\frac {137 x^5}{7200}-\frac {25 x^4}{288}+\frac {11 x^3}{36}-\frac {3 x^2}{4}+x}{x}+\frac {\left (-\frac {x^7}{5040}+\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) \log (x)}{x}\right ) \]