4.27.8 \(-a x^{k-1} y(x)+a x^k y'(x)+y''(x)=0\)

ODE
\[ -a x^{k-1} y(x)+a x^k y'(x)+y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0477762 (sec), leaf count = 90

\[\left \{\left \{y(x)\to \frac {\left (\frac {1}{k}+1\right )^{-\frac {1}{k+1}} k^{-\frac {1}{k+1}} a^{\frac {1}{k+1}} \left (x^k\right )^{\frac {1}{k}} \left (c_2 (-1)^{\frac {1}{k+1}} (k+1)-c_1 \Gamma \left (-\frac {1}{k+1},0,\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )\right )}{k+1}\right \}\right \}\]

Maple ✓
cpu = 0.248 (sec), leaf count = 133

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,x+{\it \_C2}\, \left ( \left ( k+1 \right ) \left ( {x}^{-{\frac {k}{2}}}a+{x}^{-{\frac {3\,k}{2}}-1}k \right ) {{\sl M}_{{\frac {-2-k}{2\,k+2}},\,{\frac {1+2\,k}{2\,k+2}}}\left ({\frac {a{x}^{k+1}}{k+1}}\right )}+{{\sl M}_{{\frac {k}{2\,k+2}},\,{\frac {1+2\,k}{2\,k+2}}}\left ({\frac {a{x}^{k+1}}{k+1}}\right )}{x}^{-{\frac {3\,k}{2}}-1}{k}^{2} \right ) {{\rm e}^{-{\frac {a{x}^{k+1}}{2\,k+2}}}} \right \} \] Mathematica raw input

DSolve[-(a*x^(-1 + k)*y[x]) + a*x^k*y'[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (a^(1 + k)^(-1)*(x^k)^k^(-1)*((-1)^(1 + k)^(-1)*(1 + k)*C[2] - C[1]*Ga
mma[-(1 + k)^(-1), 0, (a*(x^k)^(1 + k^(-1)))/(1 + k)]))/((1 + k^(-1))^(1 + k)^(-
1)*k^(1 + k)^(-1)*(1 + k))}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*x^k*diff(y(x),x)-a*x^(k-1)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*x+_C2*((k+1)*(x^(-1/2*k)*a+x^(-3/2*k-1)*k)*WhittakerM((-2-k)/(2*k+2),
(1+2*k)/(2*k+2),x^(k+1)/(k+1)*a)+WhittakerM(k/(2*k+2),(1+2*k)/(2*k+2),x^(k+1)/(k
+1)*a)*x^(-3/2*k-1)*k^2)*exp(-a*x^(k+1)/(2*k+2))