Internal problem ID [1417]
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
III. Exercises 7.7. Page 389
Problem number: 1.
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 }-3 y^{\prime } x +\left (4 x +3\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 61
Order:=6; dsolve(x^2*diff(y(x),x$2)-3*x*diff(y(x),x)+(3+4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x \left (\left (1-\frac {4}{3} x +\frac {2}{3} x^{2}-\frac {8}{45} x^{3}+\frac {4}{135} x^{4}-\frac {16}{4725} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{1} x^{2}+c_{2} \left (\ln \relax (x ) \left (16 x^{2}-\frac {64}{3} x^{3}+\frac {32}{3} x^{4}-\frac {128}{45} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2-8 x +\frac {256}{9} x^{3}-\frac {200}{9} x^{4}+\frac {5024}{675} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 87
AsymptoticDSolveValue[x^2*y''[x]-3*x*y'[x]+(3+4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{9} x \left (124 x^4-176 x^3+36 x^2+36 x+9\right )-\frac {8}{3} x^3 \left (2 x^2-4 x+3\right ) \log (x)\right )+c_2 \left (\frac {4 x^7}{135}-\frac {8 x^6}{45}+\frac {2 x^5}{3}-\frac {4 x^4}{3}+x^3\right ) \]