Internal problem ID [4199]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XIV. page 177
Problem number: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (-n +1\right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (-n +1\right ) x 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: 471
Order:=6; dsolve(x^2*(1-4*x)*diff(y(x),x$2)+((1-n)*x-(6-4*n)*x^2)*diff(y(x),x)+n*(1-n)*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{n} c_{1} \left (1+n x +\frac {1}{2} n \left (n +3\right ) x^{2}+\frac {1}{6} \left (n +5\right ) \left (n +4\right ) n x^{3}+\frac {1}{24} n \left (n +5\right ) \left (n +7\right ) \left (n +6\right ) x^{4}+\frac {1}{120} \left (n +9\right ) \left (n +8\right ) \left (n +7\right ) \left (n +6\right ) n x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-n x +\frac {1}{2} n \left (-3+n \right ) x^{2}-\frac {1}{6} \left (n -4\right ) \left (n -5\right ) n x^{3}+\frac {1}{24} n \left (n -5\right ) \left (n -6\right ) \left (n -7\right ) x^{4}-\frac {1}{120} \left (n -6\right ) \left (n -7\right ) \left (n -8\right ) \left (n -9\right ) n x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 2114
AsymptoticDSolveValue[x^2*(1-4*x)*y''[x]+((1-n)*x-(6-4*n)*x^2)*y'[x]+n*(1-n)*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {\left (512 n-256 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (64 \left (n-n^2\right )-128 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (16 \left (n-n^2\right )-32 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}-\frac {\left (4 \left (n-n^2\right )-8 (n+3)\right ) \left (32 n-16 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (4 \left (n-n^2\right )-8 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (-n^2+n-2 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}\right )}{(1-n) (n+3)+(n+2) (n+3)}-\frac {\left (-n^2+n-2 (n+4)\right ) \left (128 n-64 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (16 \left (n-n^2\right )-32 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (4 \left (n-n^2\right )-8 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}-\frac {\left (-n^2+n-2 (n+3)\right ) \left (32 n-16 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (4 \left (n-n^2\right )-8 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (-n^2+n-2 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}\right )}{(1-n) (n+3)+(n+2) (n+3)}\right )}{(1-n) (n+4)+(n+3) (n+4)}\right ) x^5}{(1-n) (n+5)+(n+4) (n+5)}+\frac {\left (128 n-64 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (16 \left (n-n^2\right )-32 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (4 \left (n-n^2\right )-8 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}-\frac {\left (-n^2+n-2 (n+3)\right ) \left (32 n-16 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (4 \left (n-n^2\right )-8 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (-n^2+n-2 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}\right )}{(1-n) (n+3)+(n+2) (n+3)}\right ) x^4}{(1-n) (n+4)+(n+3) (n+4)}+\frac {\left (32 n-16 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (4 \left (n-n^2\right )-8 (n+1)\right )}{(1-n) (n+1)+n (n+1)}-\frac {\left (-n^2+n-2 (n+2)\right ) \left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right )}{(1-n) (n+2)+(n+1) (n+2)}\right ) x^3}{(1-n) (n+3)+(n+2) (n+3)}+\frac {\left (8 n-4 \left (n-n^2\right )-\frac {\left (n^2+n\right ) \left (-n^2+n-2 (n+1)\right )}{(1-n) (n+1)+n (n+1)}\right ) x^2}{(1-n) (n+2)+(n+1) (n+2)}+\frac {\left (n^2+n\right ) x}{(1-n) (n+1)+n (n+1)}+1\right ) c_1 x^n+\left (\frac {\left (-256 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (64 \left (n-n^2\right )-128\right )}{1-n}-\frac {\left (16 \left (n-n^2\right )-64\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}-\frac {\left (4 \left (n-n^2\right )-24\right ) \left (-16 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (4 \left (n-n^2\right )-8\right )}{1-n}-\frac {\left (-n^2+n-4\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}\right )}{3 (1-n)+6}-\frac {\left (-n^2+n-8\right ) \left (-64 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (16 \left (n-n^2\right )-32\right )}{1-n}-\frac {\left (4 \left (n-n^2\right )-16\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}-\frac {\left (-n^2+n-6\right ) \left (-16 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (4 \left (n-n^2\right )-8\right )}{1-n}-\frac {\left (-n^2+n-4\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}\right )}{3 (1-n)+6}\right )}{4 (1-n)+12}\right ) x^5}{5 (1-n)+20}+\frac {\left (-64 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (16 \left (n-n^2\right )-32\right )}{1-n}-\frac {\left (4 \left (n-n^2\right )-16\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}-\frac {\left (-n^2+n-6\right ) \left (-16 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (4 \left (n-n^2\right )-8\right )}{1-n}-\frac {\left (-n^2+n-4\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}\right )}{3 (1-n)+6}\right ) x^4}{4 (1-n)+12}+\frac {\left (-16 \left (n-n^2\right )-\frac {\left (n^2-n\right ) \left (4 \left (n-n^2\right )-8\right )}{1-n}-\frac {\left (-n^2+n-4\right ) \left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right )}{2 (1-n)+2}\right ) x^3}{3 (1-n)+6}+\frac {\left (-4 \left (n-n^2\right )-\frac {\left (-n^2+n-2\right ) \left (n^2-n\right )}{1-n}\right ) x^2}{2 (1-n)+2}+\frac {\left (n^2-n\right ) x}{1-n}+1\right ) c_2 \]