Internal problem ID [6240]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 x^{2} y^{\prime \prime }-15 x y^{\prime }+7 \left (1+x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 77
Order:=8; dsolve(9*x^2*diff(y(x),x$2)-15*x*diff(y(x),x)+7*(x+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{\frac {1}{3}} \left (x^{2} \left (1-\frac {7}{27} x +\frac {49}{1944} x^{2}-\frac {343}{262440} x^{3}+\frac {2401}{56687040} x^{4}-\frac {2401}{2550916800} x^{5}+\frac {16807}{1101996057600} x^{6}-\frac {16807}{89261680665600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{1}+c_{2} \left (\ln \relax (x ) \left (\frac {49}{81} x^{2}-\frac {343}{2187} x^{3}+\frac {2401}{157464} x^{4}-\frac {16807}{21257640} x^{5}+\frac {117649}{4591650240} x^{6}-\frac {117649}{206624260800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-2-\frac {14}{9} x +\frac {1372}{6561} x^{3}-\frac {60025}{1889568} x^{4}+\frac {2638699}{1275458400} x^{5}-\frac {10706059}{137749507200} x^{6}+\frac {11916163}{6198727824000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.092 (sec). Leaf size: 141
AsymptoticDSolveValue[9*x^2*y''[x]-15*x*y'[x]+7*(x+1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {16807 x^{25/3}}{1101996057600}-\frac {2401 x^{22/3}}{2550916800}+\frac {2401 x^{19/3}}{56687040}-\frac {343 x^{16/3}}{262440}+\frac {49 x^{13/3}}{1944}-\frac {7 x^{10/3}}{27}+x^{7/3}\right )+c_1 \left (\frac {\sqrt [3]{x} \left (6235397 x^6-169717086 x^5+2713009950 x^4-19803722400 x^3+20832487200 x^2+107138505600 x+137749507200\right )}{137749507200}-\frac {49 x^{7/3} \left (2401 x^4-74088 x^3+1428840 x^2-14696640 x+56687040\right ) \log (x)}{9183300480}\right ) \]