Internal problem ID [6267]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 19.
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 }+3 y^{\prime } x^{2}+\left (1+3 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 81
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+3*x^2*diff(y(x),x)+(1+3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {9}{8} x +\frac {135}{256} x^{2}-\frac {315}{2048} x^{3}+\frac {8505}{262144} x^{4}-\frac {56133}{10485760} x^{5}+\frac {243243}{335544320} x^{6}-\frac {312741}{3758096384} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\frac {3}{2} x -\frac {261}{256} x^{2}+\frac {729}{2048} x^{3}-\frac {44091}{524288} x^{4}+\frac {63099}{4194304} x^{5}-\frac {1454463}{671088640} x^{6}+\frac {1403811}{5368709120} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 176
AsymptoticDSolveValue[4*x^2*y''[x]+3*x^2*y'[x]+(1+3*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {312741 x^7}{3758096384}+\frac {243243 x^6}{335544320}-\frac {56133 x^5}{10485760}+\frac {8505 x^4}{262144}-\frac {315 x^3}{2048}+\frac {135 x^2}{256}-\frac {9 x}{8}+1\right )+c_2 \left (\sqrt {x} \left (\frac {1403811 x^7}{5368709120}-\frac {1454463 x^6}{671088640}+\frac {63099 x^5}{4194304}-\frac {44091 x^4}{524288}+\frac {729 x^3}{2048}-\frac {261 x^2}{256}+\frac {3 x}{2}\right )+\sqrt {x} \left (-\frac {312741 x^7}{3758096384}+\frac {243243 x^6}{335544320}-\frac {56133 x^5}{10485760}+\frac {8505 x^4}{262144}-\frac {315 x^3}{2048}+\frac {135 x^2}{256}-\frac {9 x}{8}+1\right ) \log (x)\right ) \]