Problem 52 - Calculus for Biology and Medicine

Question

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=t^{2}\left(1-t^{2}\right), t \geq 0 $$

Step-by-Step Solution

Answer

The general solution is \( y(t) = \frac{t^3}{3} - \frac{t^5}{5} + C \).

Step 1: Understand the form of the equation

This is a first-order ordinary differential equation of the form \( \frac{dy}{dt} = f(t) \). The general solution can be found by integrating the function \( f(t) \).

Step 2: Identify the integration function

The function we need to integrate is given by \( f(t) = t^2(1-t^2) \). Simplify this to \( t^2 - t^4 \) before integrating.

Step 3: Integrate the function

Integrate the function \( t^2 - t^4 \) with respect to \( t \). Calculate the antiderivative: \[ \int (t^2 - t^4) \, dt = \frac{t^3}{3} - \frac{t^5}{5} + C \] where \( C \) is the constant of integration.

Step 4: Write the general solution

The general solution of the differential equation is \( y(t) = \frac{t^3}{3} - \frac{t^5}{5} + C \). This is the expression for \( y \) as a function of \( t \).

Key Concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that contain functions of one independent variable and its derivatives. The term 'ordinary' is used to distinguish these equations from partial differential equations, which involve multiple independent variables. In ODEs, the primary goal is to find the function or functions that satisfy the equation.
ODEs can be of various orders. The order of an ODE is determined by the highest derivative present in the equation. They can be as simple as first-order equations (which involve the first derivative) or higher order (involving second, third, or higher-order derivatives).
While higher-order ODEs exist, many foundational concepts come from understanding first-order differential equations before moving to more complex cases. In essence, solving an ODE involves integrating one or more times, depending on the order of the equation.

Integration

Integration is a fundamental concept in calculus and is used extensively in solving differential equations. It refers to the process of finding a function (the antiderivative) whose derivative is the given function. For ODEs, this process helps in finding solutions to the equations.
In the exercise, integrating the function \( f(t) = t^2(1-t^2) \) involves finding the antiderivative of the expression \( t^2 - t^4 \). This is done using basic rules of integration:

• For \( t^n \), the integral is \( \frac{t^{n+1}}{n+1} \).

Using these rules, the antiderivative of \( t^2 \) is \( \frac{t^3}{3} \), and that of \( -t^4 \) is \( -\frac{t^5}{5} \).
Once these are calculated, we add a constant of integration \( C \), leading to the solution \( y(t) = \frac{t^3}{3} - \frac{t^5}{5} + C \). This constant is essential because it represents the family of all solutions to the differential equation, reflecting different possible initial conditions.

First-order Differential Equation

First-order differential equations involve the first derivative of a function but no higher derivatives. They are often in the form \( \frac{dy}{dt} = f(t) \), which is what we see in the given exercise. Solving first-order ODEs generally involves integrating the function on the right-hand side of the equation.
The goal is to find a function \( y(t) \) that satisfies this relation.

• The equation \( \frac{dy}{dt} = t^2(1-t^2) \) is an example where \( f(t) \) is a polynomial expression.

By integrating \( t^2(1-t^2) \) as described earlier, we find its antiderivative, which offers the general solution for \( y(t) \).
First-order equations are simpler compared to higher-order ones because they involve only one integration process. However, understanding their solutions is crucial as they form the basis for more complex analyses of systems modeled by differential equations in various fields such as physics, biology, and engineering.