Internal problem ID [6231]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+2 x \left (-x +2\right ) y^{\prime }-\left (3 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 79
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+2*x*(2-x)*diff(y(x),x)-(1+3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {1}{16} x^{3}+\frac {1}{96} x^{4}+\frac {1}{768} x^{5}+\frac {1}{7680} x^{6}+\frac {1}{92160} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1+\frac {1}{2} x -\frac {1}{32} x^{3}-\frac {5}{576} x^{4}-\frac {13}{9216} x^{5}-\frac {77}{460800} x^{6}-\frac {29}{1843200} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 141
AsymptoticDSolveValue[4*x^2*y''[x]+2*x*(2-x)*y'[x]-(1+3*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^{13/2}}{46080}+\frac {x^{11/2}}{3840}+\frac {x^{9/2}}{384}+\frac {x^{7/2}}{48}+\frac {x^{5/2}}{8}+\frac {x^{3/2}}{2}+\sqrt {x}\right )+c_1 \left (\frac {\sqrt {x} \left (x^5+10 x^4+80 x^3+480 x^2+1920 x+3840\right ) \log (x)}{7680}-\frac {137 x^6+1250 x^5+8800 x^4+43200 x^3+115200 x^2-460800}{460800 \sqrt {x}}\right ) \]