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 = 0.475826 (sec), leaf count = 179
\[\left \{\left \{y(x)\to \frac {\left (8 a^2-4 a (\text {a0}-3 x)-\text {a0}^2-6 \text {a0} x+3 x^2\right ) \left (\frac {1}{9} c_2 \left (e^{-\frac {4 (a-\text {a0})}{\text {a0}+x}} \left (x-\frac {8 (a-\text {a0})^2 (6 a+\text {a0}+7 x)}{8 a^2-4 a (\text {a0}-3 x)-\text {a0}^2-6 \text {a0} x+3 x^2}\right )+12 a \text {Ei}\left (-\frac {4 (a-\text {a0})}{\text {a0}+x}\right )-12 \text {a0} \text {Ei}\left (-\frac {4 (a-\text {a0})}{\text {a0}+x}\right )+(6 a-5 \text {a0}) e^{-\frac {4 (a-\text {a0})}{\text {a0}+x}}\right )+c_1\right )}{8 a^2-4 a \text {a0}-\text {a0}^2}\right \}\right \}\]
Maple ✓
cpu = 0.127 (sec), leaf count = 132
\[ \left \{ y \left ( x \right ) =-32\, \left ( -1/8\,{{\it a0}}^{2}+ \left ( -a/2-3/4\,x \right ) {\it a0}+{a}^{2}+3/2\,ax+3/8\,{x}^{2} \right ) \left ( a-{\it a0} \right ) {\it \_C2}\,{\it Ei} \left ( 1,{\frac {4\,a-4\,{\it a0}}{{\it a0}+x}} \right ) +8\, \left ( {\it a0}+x \right ) {\it \_C2}\, \left ( -1/8\,{{\it a0}}^{2}+ \left ( -3/4\,a-x \right ) {\it a0}+{a}^{2}+5/4\,ax+1/8\,{x}^{2} \right ) {{\rm e}^{{\frac {-4\,a+4\,{\it a0}}{{\it a0}+x}}}}+{\frac {8\,{\it \_C1}}{3} \left ( -{\frac {{{\it a0}}^{2}}{8}}+ \left ( -{\frac {a}{2}}-{\frac {3\,x}{4}} \right ) {\it a0}+{a}^{2}+{\frac {3\,ax}{2}}+{\frac {3\,{x}^{2}}{8}} \right ) } \right \} \] 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] -> ((8*a^2 - a0^2 - 4*a*(a0 - 3*x) - 6*a0*x + 3*x^2)*(C[1] + (C[2]*((6*a - 5*a0)/E^((4*(a - a0))/(a0 + x)) + (x - (8*(a - a0)^2*(6*a + a0 + 7*x))/(8*a^2 - a0^2 - 4*a*(a0 - 3*x) - 6*a0*x + 3*x^2))/E^((4*(a - a0))/(a0 + x)) + 12*a*ExpIntegralEi[(-4*(a - a0))/(a0 + x)] - 12*a0*ExpIntegralEi[(-4*(a - a0))/(a0 + x)]))/9))/(8*a^2 - 4*a*a0 - a0^2)}}
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),'implicit')
Maple raw output
y(x) = -32*(-1/8*a0^2+(-1/2*a-3/4*x)*a0+a^2+3/2*a*x+3/8*x^2)*(a-a0)*_C2*Ei(1,(4*a-4*a0)/(a0+x))+8*(a0+x)*_C2*(-1/8*a0^2+(-3/4*a-x)*a0+a^2+5/4*a*x+1/8*x^2)*exp((-4*a+4*a0)/(a0+x))+8/3*_C1*(-1/8*a0^2+(-1/2*a-3/4*x)*a0+a^2+3/2*a*x+3/8*x^2)