ODE
\[ -4 (a+x) y'(x)+(\text {a0}+x)^2 y''(x)+6 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.07404 (sec), leaf count = 322
\[\left \{\left \{y(x)\to \frac {1}{3} e^{-\frac {4 a}{\text {a0}+x}} \left (12 c_2 (a-\text {a0}) e^{\frac {4 a}{\text {a0}+x}} \left (8 a^2-4 a (\text {a0}-3 x)-\text {a0}^2-6 \text {a0} x+3 x^2\right ) \text {Ei}\left (-\frac {4 (a-\text {a0})}{\text {a0}+x}\right )+8 a^2 \left (c_1 e^{\frac {4 a}{\text {a0}+x}}+3 c_2 e^{\frac {4 \text {a0}}{\text {a0}+x}} (\text {a0}+x)\right )-\text {a0}^2 c_1 e^{\frac {4 a}{\text {a0}+x}}-2 a \left (2 \text {a0} \left (c_1 e^{\frac {4 a}{\text {a0}+x}}-3 c_2 x e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )-3 x \left (2 c_1 e^{\frac {4 a}{\text {a0}+x}}+5 c_2 x e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )+9 \text {a0}^2 c_2 e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )+3 c_1 x^2 e^{\frac {4 a}{\text {a0}+x}}-6 \text {a0} c_1 x e^{\frac {4 a}{\text {a0}+x}}-3 \text {a0}^3 c_2 e^{\frac {4 \text {a0}}{\text {a0}+x}}-27 \text {a0}^2 c_2 x e^{\frac {4 \text {a0}}{\text {a0}+x}}+3 c_2 x^3 e^{\frac {4 \text {a0}}{\text {a0}+x}}-21 \text {a0} c_2 x^2 e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.151 (sec), leaf count = 133
\[\left [y \left (x \right ) = \textit {\_C1} \left (-\frac {\mathit {a0}^{2}}{3}+\frac {\left (-4 a -6 x \right ) \mathit {a0}}{3}+\frac {8 a^{2}}{3}+4 a x +x^{2}\right )+\textit {\_C2} \left (-32 \left (-\frac {\mathit {a0}^{2}}{8}+\left (-\frac {a}{2}-\frac {3 x}{4}\right ) \mathit {a0} +a^{2}+\frac {3 a x}{2}+\frac {3 x^{2}}{8}\right ) \left (a -\mathit {a0} \right ) \expIntegral \left (1, \frac {4 a -4 \mathit {a0}}{\mathit {a0} +x}\right )+8 \,{\mathrm e}^{\frac {-4 a +4 \mathit {a0}}{\mathit {a0} +x}} \left (\mathit {a0} +x \right ) \left (-\frac {\mathit {a0}^{2}}{8}+\left (-\frac {3 a}{4}-x \right ) \mathit {a0} +a^{2}+\frac {5 a x}{4}+\frac {x^{2}}{8}\right )\right )\right ]\] Mathematica raw input
DSolve[6*y[x] - 4*(a + x)*y'[x] + (a0 + x)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-(a0^2*E^((4*a)/(a0 + x))*C[1]) - 6*a0*E^((4*a)/(a0 + x))*x*C[1] + 3*E^((4*a)/(a0 + x))*x^2*C[1] - 3*a0^3*E^((4*a0)/(a0 + x))*C[2] - 27*a0^2*E^((4*a0)/(a0 + x))*x*C[2] - 21*a0*E^((4*a0)/(a0 + x))*x^2*C[2] + 3*E^((4*a0)/(a0 + x))*x^3*C[2] + 8*a^2*(E^((4*a)/(a0 + x))*C[1] + 3*E^((4*a0)/(a0 + x))*(a0 + x)*C[2]) - 2*a*(9*a0^2*E^((4*a0)/(a0 + x))*C[2] + 2*a0*(E^((4*a)/(a0 + x))*C[1] - 3*E^((4*a0)/(a0 + x))*x*C[2]) - 3*x*(2*E^((4*a)/(a0 + x))*C[1] + 5*E^((4*a0)/(a0 + x))*x*C[2])) + 12*(a - a0)*E^((4*a)/(a0 + x))*(8*a^2 - a0^2 - 4*a*(a0 - 3*x) - 6*a0*x + 3*x^2)*C[2]*ExpIntegralEi[(-4*(a - a0))/(a0 + x)])/(3*E^((4*a)/(a0 + x)))}}
Maple raw input
dsolve((a0+x)^2*diff(diff(y(x),x),x)-4*(a+x)*diff(y(x),x)+6*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(-1/3*a0^2+1/3*(-4*a-6*x)*a0+8/3*a^2+4*a*x+x^2)+_C2*(-32*(-1/8*a0^2+(-1/2*a-3/4*x)*a0+a^2+3/2*a*x+3/8*x^2)*(a-a0)*Ei(1,(4*a-4*a0)/(a0+x))+8*exp((-4*a+4*a0)/(a0+x))*(a0+x)*(-1/8*a0^2+(-3/4*a-x)*a0+a^2+5/4*a*x+1/8*x^2))]