\[ \boxed { 2\,{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -3\, \left ( y \left ( x \right ) \right ) ^{2}-4\,ay \left ( x \right ) -b-c{{\rm e}^{-2\,ax}}=0} \]
Mathematica: cpu = 0.287036 (sec), leaf count = 2831 \[ \left \{\left \{y(x)\to -\frac {2 \left (-2^{-\frac {a \sqrt {4 a^2-3 b}-2 a^2}{a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}+1} 3^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} a^{-\frac {a \sqrt {4 a^2-3 b}-2 a^2}{2 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1} b^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-1} \left (\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}\right ) c^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} e^{-2 a x} J_{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right ) \Gamma \left (\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-1}-2^{-\frac {a \sqrt {4 a^2-3 b}-2 a^2}{a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}-2} 3^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}+\frac {1}{2}} a^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}-\frac {a \sqrt {4 a^2-3 b}-2 a^2}{2 a^2}} b^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-\frac {1}{2}} c^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}+\frac {1}{2}} e^{-2 a x} \left (J_{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}-1}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right )-J_{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right )\right ) \Gamma \left (\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-\frac {1}{2}}+c_1 \left (-2^{-\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}-2} 3^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}+\frac {1}{2}} b^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-\frac {1}{2}} c^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}+\frac {1}{2}} e^{-2 a x} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-\frac {1}{2}} \left (J_{-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}-1}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right )-J_{1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right )\right ) \Gamma \left (1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}\right ) a^{-\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{2 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}-2^{-\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}+1} 3^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} b^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-1} \left (\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}\right ) c^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} e^{-2 a x} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}-1} J_{-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right ) \Gamma \left (1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}\right ) a^{-\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{2 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1}\right )\right )}{3 \left (2^{-\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}} 3^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} b^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} c^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{4 a^2}+\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} J_{-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right ) c_1 \Gamma \left (1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}\right ) a^{-\frac {-2 a^2-\sqrt {4 a^2-3 b} a}{2 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}+2^{\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}-\frac {a \sqrt {4 a^2-3 b}-2 a^2}{a^2}} 3^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} b^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} c^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a \sqrt {4 a^2-3 b}-2 a^2}{4 a^2}-\frac {\sqrt {4 a^4-3 a^2 b}}{4 a^2}} J_{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{2 a}\right ) \Gamma \left (\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1\right ) a^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}-\frac {a \sqrt {4 a^2-3 b}-2 a^2}{2 a^2}}\right )}\right \}\right \} \]
Maple: cpu = 0.156 (sec), leaf count = 420 \[ \left \{ y \left ( x \right ) = \left ( -{\frac {\sqrt {3}{\it \_C1}}{3} \sqrt {c}{{\sl Y}_{-{\frac {1}{2\,a} \left ( \sqrt {4\,{a}^{2}-3\,b}-2 \,a \right ) }}\left ({\frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}} \right )} \left ( {{\sl Y}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^{2}-3\,b}} }\left ({\frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}}\right )}{\it \_C1}+{{\sl J}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^{2}-3\,b}}}\left ({ \frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}}\right )} \right ) ^{-1} }-{\frac {\sqrt {3}}{3}\sqrt {c}{{\sl J}_{-{\frac {1}{2\,a} \left ( \sqrt {4\,{a}^{2}-3\,b}-2\,a \right ) }}\left ({\frac {\sqrt {3}{{\rm e} ^{-ax}}}{2\,a}\sqrt {c}}\right )} \left ( {{\sl Y}_{-{\frac {1}{2\,a} \sqrt {4\,{a}^{2}-3\,b}}}\left ({\frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a} \sqrt {c}}\right )}{\it \_C1}+{{\sl J}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^ {2}-3\,b}}}\left ({\frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}} \right )} \right ) ^{-1}} \right ) {{\rm e}^{-ax}}-{\frac {1}{3} \left ( \left ( \sqrt {4\,{a}^{2}-3\,b}{\it \_C1}+2\,{\it \_C1}\,a \right ) { {\sl Y}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^{2}-3\,b}}}\left ({\frac { \sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}}\right )}+ \left ( \sqrt {4\,{a }^{2}-3\,b}+2\,a \right ) {{\sl J}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^{2}- 3\,b}}}\left ({\frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}}\right )} \right ) \left ( {{\sl Y}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^{2}-3\,b}} }\left ({\frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}}\right )}{\it \_C1}+{{\sl J}_{-{\frac {1}{2\,a}\sqrt {4\,{a}^{2}-3\,b}}}\left ({ \frac {\sqrt {3}{{\rm e}^{-ax}}}{2\,a}\sqrt {c}}\right )} \right ) ^{-1} } \right \} \]