Internal problem ID [1377]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
II. Exercises 7.6. Page 374
Problem number: 25.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (4 x +1\right ) y^{\prime \prime }-x \left (-4 x +1\right ) y^{\prime }+\left (x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 75
Order:=8; dsolve(x^2*(1+4*x)*diff(y(x),x$2)-x*(1-4*x)*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-5 x +\frac {85}{4} x^{2}-\frac {3145}{36} x^{3}+\frac {204425}{576} x^{4}-\frac {825877}{576} x^{5}+\frac {119752165}{20736} x^{6}-\frac {23591176505}{1016064} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (2 x -\frac {39}{4} x^{2}+\frac {4499}{108} x^{3}-\frac {594305}{3456} x^{4}+\frac {2420617}{3456} x^{5}-\frac {117547073}{41472} x^{6}+\frac {162576422327}{14224896} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 158
AsymptoticDSolveValue[x^2*(1+4*x)*y''[x]-x*(1-4*x)*y'[x]+(1+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (-\frac {23591176505 x^7}{1016064}+\frac {119752165 x^6}{20736}-\frac {825877 x^5}{576}+\frac {204425 x^4}{576}-\frac {3145 x^3}{36}+\frac {85 x^2}{4}-5 x+1\right )+c_2 \left (x \left (\frac {162576422327 x^7}{14224896}-\frac {117547073 x^6}{41472}+\frac {2420617 x^5}{3456}-\frac {594305 x^4}{3456}+\frac {4499 x^3}{108}-\frac {39 x^2}{4}+2 x\right )+x \left (-\frac {23591176505 x^7}{1016064}+\frac {119752165 x^6}{20736}-\frac {825877 x^5}{576}+\frac {204425 x^4}{576}-\frac {3145 x^3}{36}+\frac {85 x^2}{4}-5 x+1\right ) \log (x)\right ) \]