Given an ODE y''+16y = 4\sin(4t) a. Find the general solution of homogeneous ODE (the left hand...
Question:
Given an ODE {eq}y''+16y = 4\sin(4t) {/eq}.
a. Find the general solution of homogeneous ODE (the left hand side) and call it yh.
b. Verify {eq}y= -0.5 t \cos (4t) {/eq} is a solution, i.e. show that {eq}y''p+16yp = 4\sin(4t) {/eq}.
c. Construct the general solution of this y = yh + yp.
Second Order Differential Equation:
The ordinary differential equation that involving second order derivative is called differential equation of order 2. The complementary function of the differential equation when the roots of auxiliary equation are complex is given by {eq}C.F. = {e^{\alpha x}}\left( {{c_1}\cos \beta x + {c_2}\sin \beta x} \right) {/eq}.
The particular integral is given by when {eq}F\left( { - {D^2}} \right) \ne 0 {/eq},
{eq}\begin{align*} P.I. &= \dfrac{1}{{F\left( D \right)}}\left( {\sin ax} \right)\\ &= I.P.\left\{ {\dfrac{1}{{F\left( D \right)}} \cdot {e^{iax}}} \right\} \end{align*} {/eq},
Where I.P. is the imaginary part.
Answer and Explanation: 1
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The differential equation is given as, {eq}y'' + 16y = 4\sin 4t {/eq}.
Let {eq}{D^2}y = y'' {/eq} then {eq}\left( {{D^2} + 16} \right)y =...
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Separation of Variables to Solve System Differential Equations
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