Internal problem ID [1370]
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: 18.
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 +9\right ) y^{\prime \prime }+3 y^{\prime } x +\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: 0.012 (sec). Leaf size: 69
Order:=6; dsolve(x^2*(9+4*x)*diff(y(x),x$2)+3*x*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{\frac {1}{3}} \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{81} x +\frac {25}{26244} x^{2}-\frac {3025}{19131876} x^{3}+\frac {874225}{24794911296} x^{4}-\frac {18498601}{2008387814976} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {14}{81} x -\frac {35}{2916} x^{2}+\frac {110495}{57395628} x^{3}-\frac {62786185}{148769467776} x^{4}+\frac {1315043653}{12050326889856} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 134
AsymptoticDSolveValue[x^2*(9+4*x)*y''[x]+3*x*y'[x]+(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {18498601 x^5}{2008387814976}+\frac {874225 x^4}{24794911296}-\frac {3025 x^3}{19131876}+\frac {25 x^2}{26244}-\frac {x}{81}+1\right )+c_2 \left (\sqrt [3]{x} \left (\frac {1315043653 x^5}{12050326889856}-\frac {62786185 x^4}{148769467776}+\frac {110495 x^3}{57395628}-\frac {35 x^2}{2916}+\frac {14 x}{81}\right )+\sqrt [3]{x} \left (-\frac {18498601 x^5}{2008387814976}+\frac {874225 x^4}{24794911296}-\frac {3025 x^3}{19131876}+\frac {25 x^2}{26244}-\frac {x}{81}+1\right ) \log (x)\right ) \]