Internal problem ID [5739]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert.
(A) Drill Exercises . Page 194
Problem number: 3(a).
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.04 (sec). Leaf size: 1219
Order:=8; dsolve(x^3*diff(y(x),x$3)+2*x^2*diff(y(x),x$2)+(x+x^2)*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{-1+i \sqrt {3}} x +\frac {1}{2} \frac {1}{\left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{2}+\frac {1}{6} \frac {1}{\left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{3}+\frac {1}{24} \frac {1}{\left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{4}+\frac {1}{120} \frac {1}{\left (i \sqrt {3}-5\right ) \left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{5}+\frac {1}{720} \frac {1}{\left (i \sqrt {3}-6\right ) \left (i \sqrt {3}-5\right ) \left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{6}+\frac {1}{5040} \frac {1}{\left (i \sqrt {3}-7\right ) \left (i \sqrt {3}-6\right ) \left (i \sqrt {3}-5\right ) \left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{1+i \sqrt {3}} x +\frac {1}{2} \frac {1}{\left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{2}-\frac {1}{6} \frac {1}{\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{3}+\frac {1}{24} \frac {1}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{4}-\frac {1}{120} \frac {1}{\left (5+i \sqrt {3}\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{5}+\frac {1}{720} \frac {1}{\left (i \sqrt {3}+6\right ) \left (5+i \sqrt {3}\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{6}-\frac {1}{5040} \frac {1}{\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+6\right ) \left (5+i \sqrt {3}\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{3} \left (1-x +\frac {1}{3} x^{2}-\frac {1}{21} x^{3}+\frac {1}{273} x^{4}-\frac {1}{5733} x^{5}+\frac {1}{177723} x^{6}-\frac {1}{7642089} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 3447
AsymptoticDSolveValue[x^3*y'''[x]+2*x^2*y''[x]+(x+x^2)*y'[x]+x*y[x]==0,y[x],{x,0,7}]
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