Question

Denote by L(t) the length of a fish at time t, and assume that the fish grows according to von Bertalanffy's equation.

(dL/dt)=k(43−L(t)) with L(0)=6, Solve the differential equation.

Answer 1

To solve the differential equation (dL/dt) = k(43 - L(t)) with L(0) = 6, we can separate the variables and integrate to find L(t) = 43 - e^(-kt + ln|37|).

Step-by-step explanation:

To solve the differential equation (dL/dt) = k(43 - L(t)) with L(0) = 6, we can separate the variables and integrate.

First, separate the variables by moving L(t) terms to one side and dt terms to the other side. We get: (1/(43 - L(t)))dL = k dt

Now, integrate both sides. Integrate the left side using the natural logarithm and the right side as a constant times t. We get: ∫(1/(43 - L(t)))dL = ∫k dt

The left side can be integrated using substitution where u = 43 - L(t) and du = - dL. After integrating, we get: -ln|43 - L(t)| = kt + C

Using the initial condition L(0) = 6, we can plug in the values and solve for the constant C. We find C = -ln|43 - 6| = -ln|37|

Substituting the value of C back into the equation, we get: -ln|43 - L(t)| = kt -ln|37|.

To simplify further, we can take the exponential of both sides: |43 - L(t)| = e^(-kt + ln|37|)

Finally, we can solve for L(t) by taking the absolute value of both sides: L(t) = 43 - e^(-kt + ln|37|).