ODE
\[ y'''(x)-a^2 y'(x)+2 a^2 y(x)-2 y''(x)=\sinh (x) \] ODE Classification
[[_3rd_order, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.249127 (sec), leaf count = 52
\[\left \{\left \{y(x)\to \frac {e^{-x}-3 e^x}{6-6 a^2}+c_1 e^{-a x}+c_3 e^{a x}+c_2 e^{2 x}\right \}\right \}\]
Maple ✓
cpu = 0.75 (sec), leaf count = 114
\[\left [y \left (x \right ) = -\frac {-6 a^{3} {\mathrm e}^{x}-2 \,{\mathrm e}^{2 x} \sinh \left (3 x \right ) a^{3}+2 \,{\mathrm e}^{2 x} \cosh \left (3 x \right ) a^{3}+24 a \,{\mathrm e}^{x}+2 \,{\mathrm e}^{2 x} \sinh \left (3 x \right ) a -2 \,{\mathrm e}^{2 x} \cosh \left (3 x \right ) a -6 a \,{\mathrm e}^{-x}}{12 a \left (a^{2}-4\right ) \left (a -1\right ) \left (1+a \right )}+{\mathrm e}^{2 x} \textit {\_C1} +\textit {\_C2} \,{\mathrm e}^{a x}+\textit {\_C3} \,{\mathrm e}^{-a x}\right ]\] Mathematica raw input
DSolve[2*a^2*y[x] - a^2*y'[x] - 2*y''[x] + y'''[x] == Sinh[x],y[x],x]
Mathematica raw output
{{y[x] -> (E^(-x) - 3*E^x)/(6 - 6*a^2) + C[1]/E^(a*x) + E^(2*x)*C[2] + E^(a*x)*C[3]}}
Maple raw input
dsolve(diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-a^2*diff(y(x),x)+2*a^2*y(x) = sinh(x), y(x))
Maple raw output
[y(x) = -1/12*(-6*a^3*exp(x)-2*exp(2*x)*sinh(3*x)*a^3+2*exp(2*x)*cosh(3*x)*a^3+24*a*exp(x)+2*exp(2*x)*sinh(3*x)*a-2*exp(2*x)*cosh(3*x)*a-6*a*exp(-x))/a/(a^2-4)/(a-1)/(1+a)+exp(2*x)*_C1+_C2*exp(a*x)+_C3*exp(-a*x)]