4.43.26 \(y'''(x)-a^2 y'(x)+2 a^2 y(x)-2 y''(x)=\sinh (x)\)

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 = 0.0943293 (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 = 0.057 (sec), leaf count = 214

\[ \left \{ y \left ( x \right ) ={\frac {1}{12\,{a}^{5}-60\,{a}^{3}+48\,a} \left ( 3\, \left ( \left ( a-2 \right ) {{\rm e}^{-ax}}+{{\rm e}^{ax}} \left ( a+2 \right ) \right ) \left ( a-1 \right ) \cosh \left ( \left ( 1+a \right ) x \right ) -3\, \left ( \left ( a-2 \right ) {{\rm e}^{-ax}}+{{\rm e}^{ax}} \left ( a+2 \right ) \right ) \left ( 1+a \right ) \cosh \left ( \left ( a-1 \right ) x \right ) -3\, \left ( \left ( 2-a \right ) {{\rm e}^{-ax}}+{{\rm e}^{ax}} \left ( a+2 \right ) \right ) \left ( a-1 \right ) \sinh \left ( \left ( 1+a \right ) x \right ) +12\, \left ( 1+a \right ) \left ( \left ( \left ( -a/4+1/2 \right ) {{\rm e}^{-ax}}+1/4\,{{\rm e}^{ax}} \left ( a+2 \right ) \right ) \sinh \left ( \left ( a-1 \right ) x \right ) +a \left ( a-1 \right ) \left ( \left ( {\it \_C3}\,{a}^{2}-4\,{\it \_C3} \right ) {{\rm e}^{-ax}}+ \left ( {\it \_C2}\,{a}^{2}-4\,{\it \_C2} \right ) {{\rm e}^{ax}}+{{\rm e}^{2\,x}} \left ( {a}^{2}{\it \_C1}+1/6\,\sinh \left ( 3\,x \right ) -4\,{\it \_C1}+1/2\,\cosh \left ( x \right ) -1/2\,\sinh \left ( x \right ) -1/6\,\cosh \left ( 3\,x \right ) \right ) \right ) \right ) \right ) } \right \} \] 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),'implicit')

Maple raw output

y(x) = (3*((a-2)*exp(-a*x)+exp(a*x)*(a+2))*(a-1)*cosh((1+a)*x)-3*((a-2)*exp(-a*x
)+exp(a*x)*(a+2))*(1+a)*cosh((a-1)*x)-3*((2-a)*exp(-a*x)+exp(a*x)*(a+2))*(a-1)*s
inh((1+a)*x)+12*(1+a)*(((-1/4*a+1/2)*exp(-a*x)+1/4*exp(a*x)*(a+2))*sinh((a-1)*x)
+a*(a-1)*((_C3*a^2-4*_C3)*exp(-a*x)+(_C2*a^2-4*_C2)*exp(a*x)+exp(2*x)*(a^2*_C1+1
/6*sinh(3*x)-4*_C1+1/2*cosh(x)-1/2*sinh(x)-1/6*cosh(3*x)))))/(12*a^5-60*a^3+48*a
)