(a) Find the solution of the differential equation (d y)/( d t)=4 y which satisfies the initial

step 1 You guys, let's do problem 2. In this problem with the we need to find the differential solution

(a) Find the solution of the differential equation (d y)/( d...

(a) Find the solution of the differential equation
$$
\frac{\mathrm{d} y}{\mathrm{~d} t}=4 y
$$
which satisfies the initial condition, $y(0)=6$.
(b) Find the solution of the differential equation
$$
\frac{\mathrm{d} y}{\mathrm{~d} t}=-5 y
$$
which satisfies the initial condition, $y(0)=2$.

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Key Concepts

First-Order Differential Equations

First-order differential equations involve only the first derivative of an unknown function and are fundamental in describing processes where the rate of change depends on the current state. They are often modeled in natural sciences and engineering to capture dynamical behavior.

Separation of Variables

This is a method used to solve differential equations by rearranging the equation so that each variable and its differential are on opposite sides of the equation. Once separated, each side can be integrated independently, leading to the general solution of the equation.

Exponential Growth and Decay

Exponential growth and decay describe processes where the rate of change of a quantity is proportional to the quantity itself. This relationship is modeled by differential equations that, when solved, yield exponential functions representing rapidly changing phenomena, either increasing (growth) or decreasing (decay) over time.

Initial Value Problems

An initial value problem specifies the value of the unknown function at a particular point, providing a starting point for solving the differential equation. This condition uniquely determines the constant of integration in the general solution, yielding a specific solution to the problem.

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