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.

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

Step-by-step 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

Integrating on both sides

Then , we get

Where C is integration constant.

Now, t replace by  t+c

Then, we get

because c+C=K

Different w.r.t then we get

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

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

step-by-step explanation: