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If x(t) is a solution to the equation x' = f(x), show that x(t + c) is also a solution, for any constant c. Is x(t) + c a solution? Explain why.

User Alqueraf
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1 Answer

3 votes

Answer:

Yes, x(t)+C is also a solution of given equation.

Explanation:

We are given that x(t) is a solution of the equation x'=f(x)

We have to show that x(t+c) is also a solution of given equation and check x(t)+c is a solution of equation.

Suppose x'=1


(dx)/(dt)=1


dx=dt

Integrating on both sides

Then , we get


x=t+c

Where C is integration constant.

Now, t replace by t+c

Then, we get


x(t+c)=t+c+c=t+K because c+C=K

Different w.r.t then we get


x'=1

Therefore, x(t+c) is also solution because it satisfied the given equation.

Now, x(t)+C=t+(c+C)=t+L where L=c+C=Constant

Differentiate w.r.t time

Then, we get
x'=1

Yes, x(t)+C is also solution of given equation because it satisfied given equation

User Kesandal
by
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