Answer to Question #311030 in Differential Equations for kxngToooch

Question

Find the general solutions of the following differential equations using D-operator methods:

(D + 4)² x = sinh 4t

Answer 1

Solution;

The homogeneous solution of the equation is;

$(D+4)^2x=0$

From which the characteristic equation is;

$(m+4)^2=0$

$m=-4,-4$

We obtain the complementary solution;

$C_1e^{-4t}+C_2e^{-4t}$

The particular integral if the equation is;

$P.I=\frac{sin4t}{(D+4)^2}$

We know that;

$sinh(an)=\frac{e^{an}-e^{-an}}{2}$

Substitute into the P.I;

$P.I=\frac{e^{4t}-e^{-4t}}{2(D+4)^2}$

Hence;

$P.I=\frac{e^{4t}}{128}-\frac{t^2e^{-4t}}{4}$

The solution is;

$f(t)=C_1e^{-4t}+tC_2e^{-4t}-\frac{t^2e^{-4t}}{4}+\frac{e^{4t}}{128}$