Internal problem ID [5741]
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(c).
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}+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: 0.093 (sec). Leaf size: 120
Order:=8; dsolve(x^3*diff(y(x),x$3)-2*x^2*diff(y(x),x$2)+(x^2+2*x)*diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{3} \left (1-\frac {1}{4} x +\frac {1}{40} x^{2}-\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}-\frac {1}{806400} x^{5}+\frac {1}{43545600} x^{6}-\frac {1}{3048192000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (720-908 x +152 x^{2}-11 x^{3}+\frac {4}{9} x^{4}-\frac {79}{7056} x^{5}+\frac {517}{2822400} x^{6}-\frac {851}{457228800} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} x^{2}+\left (-24-12 x -6 x^{2}+\frac {5}{8} x^{4}-\frac {39}{400} x^{5}+\frac {49}{7200} x^{6}-\frac {199}{705600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{3}+\ln \relax (x ) \left (x^{2} c_{2} \left (\left (-240\right ) x +60 x^{2}-6 x^{3}+\frac {1}{3} x^{4}-\frac {1}{84} x^{5}+\frac {1}{3360} x^{6}-\frac {1}{181440} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+2 c_{3} \left (x^{3}-\frac {1}{4} x^{4}+\frac {1}{40} x^{5}-\frac {1}{720} x^{6}+\frac {1}{20160} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 1.031 (sec). Leaf size: 186
AsymptoticDSolveValue[x^3*y'''[x]-2*x^2*y''[x]+(x^2+2*x)*y'[x]-x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {\left (x^3-18 x^2+180 x-720\right ) x^3 \log (x)}{4320}+\frac {-167 x^6+2466 x^5-17100 x^4+14400 x^3+129600 x^2+259200 x+518400}{259200}\right )+c_2 \left (\frac {x^3 \left (x^5-40 x^4+1120 x^3-20160 x^2+201600 x-806400\right ) \log (x)}{2419200}-\frac {x^2 \left (2941 x^6-106720 x^5+2618560 x^4-38666880 x^3+268128000 x^2-225792000 x-2032128000\right )}{2032128000}\right )+c_3 \left (\frac {x^9}{43545600}-\frac {x^8}{806400}+\frac {x^7}{20160}-\frac {x^6}{720}+\frac {x^5}{40}-\frac {x^4}{4}+x^3\right ) \]