The three curves plotted below are associated with the solution to a non-homogen

Question

The three curves plotted below are associated with the solution to a non-homogeneous, 2^nd order, constant coefficient linear ODE. Match each of the three descriptions to the appropriate curve in the graph. The complete solution to the initial value problem The transient component of the solution to the initial value problem The steady state component of the solution to the initial value problem (Note that curve A merges into curve C) Explain why you matched the descriptions and curves the way you did.

Explanation / Answer

(a)

second order linear ODE with contant coefficient is in the formatio is: a d2y/dx2+b dy/dx+cy=f(x).

the above equation we can also write as ay2+by1+cy=f(x)

now i am sloving the equation for the initial values without undetermined coefficients

ay2+by1+cy=f(x) here, let we can say h is a homogeneous and j is particular solution

ay2+by1+cy=0

a(h2+j2)+b(h1+j1)+c(h+j)

ah2+bh1+ch+aj2+bj1+cj from this we can get the initial values.

(b)

when we matched the curves we can get the different vlues.so, first we calculate the initial values of the curves and then find the second order values.The merging of curves A,B are give the same value after merge but they give different values before merging of two curves.

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