Question

Solve the following differential equations (show your solutions):
a. dx/dt=r , x(0)=a
b. dx/dt=rx , x(0)=a
c. dx/dt=rx(1-x) , x(0)=a

Answer 1

All of them are separable.
For the third one, you require partial fractions.

Answer 2

solution please?

Answer 3

\[\frac{dx}{dt}=r\\
dx=r\ dt\\
x=rt+C\\
a=r(0)+C=C \Rightarrow x=rt+a\]

Answer 4

how about letter c?

Answer 5

\[\frac{dx}{dt}=rx\\
\frac{dx}{x}=r\ dt\\
\ln|x|=rt+C\\
\ln|a|=r(0)+C=C\Rightarrow \ln|x|=rt+\ln|a|\\
\ln|x|-\ln|a|=rt\\
\left |\frac{x}{a}\right |=e^{rt}\\
|x|=|a|e^{rt}\]