Question

In Problem 21 a one-parameter family of solutions of the DE$P\text{'}=P(1-P)$ is given. Does any solution curve pass through the point (0,3) Through the point(0,1)?

Short answer

There exists a solution curve passing through $(0,3)$ but not through $(0,1)$.

Step-by-step solution

Definition of differential equation.

A differential equation is defined as the derivative function of one or more unknown functions (dependable variables) with respect to one or more undependable variables.

Family of solutions.

The family of solutions is given as:

$P=\frac{c_1{e}^t}{1+c_1{e}^t}$.....(1)

Substitute$t=0$and$P=3$into (1). If you can find$c_1$that will satisfy that solution, then there exist a solution curve passing through$(0,3)$.

$$\begin{gathered} 3=\frac{c_1{e}^0}{1+c_1{e}^0} \\ 3=\frac{c_1}{1+c_1} \end{gathered}$$

Multiply$1+c_1$on both sides,

$$\begin{gathered} 3(1+c_1)=c_1 \\ 3+3c_1=c_1 \end{gathered}$$

Simplify,

$$\begin{gathered} 2c_1=-3 \\ {c}_1=-\frac{3}{2} \end{gathered}$$

Since$c_1$exists, there exists a solution curve passing through$(0,1)$,

Family of solutions.

Further, substitute $t=0$and$P=1$into (1):

$$\begin{gathered} 1=\frac{c_1{e}^0}{1+c_1{e}^0} \\ 1=\frac{c_1}{1+c_1} \end{gathered}$$

Multiply$1+c_1$on both sides,

$$\begin{gathered} (1+c_1)=c_1 \\ 1=0 \end{gathered}$$

Which is not possible, therefore, there does not exist a solution curve passing through$(0,1)$.

There exists a solution curve passing through$(0,3)$but not through$(0,1)$.