Internal problem ID [6238]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-5 x y^{\prime }+\left (8+5 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 77
Order:=8; dsolve(x^2*diff(y(x),x$2)-5*x*diff(y(x),x)+(8+5*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (1-\frac {5}{3} x +\frac {25}{24} x^{2}-\frac {25}{72} x^{3}+\frac {125}{1728} x^{4}-\frac {125}{12096} x^{5}+\frac {625}{580608} x^{6}-\frac {3125}{36578304} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{1} x^{2}+c_{2} \left (\ln \relax (x ) \left (25 x^{2}-\frac {125}{3} x^{3}+\frac {625}{24} x^{4}-\frac {625}{72} x^{5}+\frac {3125}{1728} x^{6}-\frac {3125}{12096} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-2-10 x +\frac {500}{9} x^{3}-\frac {15625}{288} x^{4}+\frac {19625}{864} x^{5}-\frac {56875}{10368} x^{6}+\frac {443125}{508032} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right )\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 123
AsymptoticDSolveValue[x^2*y''[x]-5*x*y'[x]+(8+5*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^2 \left (33125 x^6-140250 x^5+348750 x^4-396000 x^3+64800 x^2+51840 x+10368\right )}{10368}-\frac {25 x^4 \left (125 x^4-600 x^3+1800 x^2-2880 x+1728\right ) \log (x)}{3456}\right )+c_2 \left (\frac {625 x^{10}}{580608}-\frac {125 x^9}{12096}+\frac {125 x^8}{1728}-\frac {25 x^7}{72}+\frac {25 x^6}{24}-\frac {5 x^5}{3}+x^4\right ) \]