\[ \left \{x'(t)=a x(t)+b y(t),y'(t)=b y(t)+c x(t)\right \} \] ✓ Mathematica : cpu = 0.0299104 (sec), leaf count = 696
DSolve[{Derivative[1][x][t] == a*x[t] + b*y[t], Derivative[1][y][t] == c*x[t] + b*y[t]},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to \frac {c_1 \left (a \left (-e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )+a e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+b e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-b e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{2 \sqrt {a^2-2 a b+b^2+4 b c}}-\frac {b c_2 \left (e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{\sqrt {a^2-2 a b+b^2+4 b c}},y(t)\to \frac {c_2 \left (a e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-a e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-b e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+b e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{2 \sqrt {a^2-2 a b+b^2+4 b c}}-\frac {c c_1 \left (e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{\sqrt {a^2-2 a b+b^2+4 b c}}\right \}\right \}\] ✓ Maple : cpu = 0.078 (sec), leaf count = 177
dsolve({diff(x(t),t) = a*x(t)+b*y(t), diff(y(t),t) = c*x(t)+b*y(t)})
\[\left \{x \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (a +b +\sqrt {b^{2}+\left (-2 a +4 c \right ) b +a^{2}}\right ) t}{2}}+c_{2} {\mathrm e}^{\frac {\left (a +b -\sqrt {b^{2}+\left (-2 a +4 c \right ) b +a^{2}}\right ) t}{2}}, y \left (t \right ) = \frac {-c_{2} \left (a -b +\sqrt {b^{2}+\left (-2 a +4 c \right ) b +a^{2}}\right ) {\mathrm e}^{\frac {\left (a +b -\sqrt {b^{2}+\left (-2 a +4 c \right ) b +a^{2}}\right ) t}{2}}-c_{1} {\mathrm e}^{\frac {\left (a +b +\sqrt {b^{2}+\left (-2 a +4 c \right ) b +a^{2}}\right ) t}{2}} \left (a -b -\sqrt {b^{2}+\left (-2 a +4 c \right ) b +a^{2}}\right )}{2 b}\right \}\]