Internal problem ID [6204]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 \left (x -4\right )^{2} y^{\prime \prime }+\left (x -4\right ) \left (x -8\right ) y^{\prime }+x y=0} \end {gather*} With the expansion point for the power series method at \(x = 4\).
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 83
Order:=8; dsolve(4*(x-4)^2*diff(y(x),x$2)+(x-4)*(x-8)*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=4);
\[ y \relax (x ) = \left (\left (\frac {3}{4} \left (x -4\right )-\frac {13}{64} \left (x -4\right )^{2}+\frac {31}{1152} \left (x -4\right )^{3}-\frac {173}{73728} \left (x -4\right )^{4}+\frac {187}{1228800} \left (x -4\right )^{5}-\frac {463}{58982400} \left (x -4\right )^{6}+\frac {971}{2890137600} \left (x -4\right )^{7}+\mathrm {O}\left (\left (x -4\right )^{8}\right )\right ) c_{2}+\left (c_{2} \ln \left (x -4\right )+c_{1}\right ) \left (1-\frac {1}{2} \left (x -4\right )+\frac {3}{32} \left (x -4\right )^{2}-\frac {1}{96} \left (x -4\right )^{3}+\frac {5}{6144} \left (x -4\right )^{4}-\frac {1}{20480} \left (x -4\right )^{5}+\frac {7}{2949120} \left (x -4\right )^{6}-\frac {1}{10321920} \left (x -4\right )^{7}+\mathrm {O}\left (\left (x -4\right )^{8}\right )\right )\right ) \left (x -4\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 222
AsymptoticDSolveValue[4*(x-4)^2*y''[x]+(x-4)*(x-8)*y'[x]+x*y[x]==0,y[x],{x,4,7}]
\[ y(x)\to c_1 \left (-\frac {(x-4)^7}{10321920}+\frac {7 (x-4)^6}{2949120}-\frac {(x-4)^5}{20480}+\frac {5 (x-4)^4}{6144}-\frac {1}{96} (x-4)^3+\frac {3}{32} (x-4)^2+\frac {4-x}{2}+1\right ) (x-4)+c_2 \left ((x-4) \left (\frac {971 (x-4)^7}{2890137600}-\frac {463 (x-4)^6}{58982400}+\frac {187 (x-4)^5}{1228800}-\frac {173 (x-4)^4}{73728}+\frac {31 (x-4)^3}{1152}-\frac {13}{64} (x-4)^2+\frac {4-x}{4}+x-4\right )+\left (-\frac {(x-4)^7}{10321920}+\frac {7 (x-4)^6}{2949120}-\frac {(x-4)^5}{20480}+\frac {5 (x-4)^4}{6144}-\frac {1}{96} (x-4)^3+\frac {3}{32} (x-4)^2+\frac {4-x}{2}+1\right ) (x-4) \log (x-4)\right ) \]