Let c be a positive number. A differential equation of the form dy dt = ky1 + c where k is a positive constant, is called a doomsday equation because the exponent in the expression ky1 + c is larger than the exponent 1 for natural growth. (a) Determine the solution that satisfies the initial condition y(0) = y0.

Differential equation is used to relate equations and their derivatives.

The solution that satisfies the initial condition is:

The differential equation is given as:

Multiply both sides by dt

Divide both sides by

Integrate both sides

Remove constants

Rewrite as:

Now, integrate both sides

This gives

At the initial condition,

So, we have:

Substitute in

Multiply through by c

Rewrite as:

Take inverse of both sides

Multiply both sides by -1

Take c-th root of both sides

Express as exponents

Express y as a function of t

Hence, the solution that satisfies the initial condition is:

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Step-by-step explanation:

We are given that  differential equation

Where k=Positive constant

c=Positive number

Initial condition

Substitute the value of initial condition

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step-by-step explanation: