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Problem 2: Application of method of variation of parameters
Report problem 2 from.

Given: A L2-ODE-CC
The non-homogeneous linear 2nd order ordinary differential equation with constant coefficients(L2-ODE-CC)

Note: There is a typo in. The coefficient of $$y$$ given in the ODE should be $$a_0^2$$ instead of $$a_0$$.

Given the initial conditions being

Find: The final solution to the L2-ODE-CC
Find the final solution:

and compare ($$)~($$) to Eqns (2.4)~(2.5) in Dong 2012

Solution: Using method of variation of parameters

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On our honors, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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First, we assume the homogeneous solution to the L2-ODE-CC is

with $$r$$ to be determined.

Instituting ($$) back to the left-hand-side of ($$) yields:

solve the characteristic equation

thus the homogenous solutions turn out to be

We take one of the homogeneous solution $$y_1=cosa_0t$$ in ($$) to form the following trial solution to the non-homogeneous L2-ODE-CC

where $$A(t)$$ is the unknown to be found.

Instituting ($$) into ($$) yeilds

Assume that

then ($$) turns to be

Note that ($$) is a L1-ODE-VC which could be solved directly or by IFM.

Noticing that

{{NumBlk|:|$$ \left\{\begin{matrix} \phi_{Bt}=-a_0sina_0t \\ \phi_{tB}=-2a_0sina_0t \end{matrix}\right.\Rightarrow \phi_{Bt}\neq \phi_{tB}$$|$$}}

i.e. ($$) is not exact and IFM should be used to solve it

Multiply ($$) with certain integration factor $$h(t)$$

Recall the 2nd exactness condition $$\bar{\phi}_{Bt}=\bar{\phi}_{tB}$$, we have:

i.e.

where $$k_1$$ is an integrating constant.

Plug ($$) back into ($$), we get the solution

where $$k_2$$ is an integrating constant.

Integrating ($$) yeilds

where $$k_3$$ is another integrating constant.

Expand ($$), we have:

Plug ($$) back into ($$), we get the final solution to ($$):

where $$C_1$$ and $$C_2$$ are two arbitrary constants.

Note the condition that $$t\geqslant t_0$$ and also integration range being $$[t_0, t]$$, the solution is modified to be

Consider two initial conditions as expressed in ($$), we have

{{NumBlk|:|$$ \left\{\begin{matrix} \Complex_1=y'_0 \\ \Complex_2=y_0 \end{matrix}\right. $$|$$}}

Therefore, the final solution to ($$) is

Compared with Eqns(2.5) in Dong 2012, we know they are in the same expression.