Problem 36 - Calculus for Biology and Medicine

Question

$$ \text { In Problems } , \text { solve each differential equation. } $$$$ \frac{d y}{d x}=(1+y)^{2} $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{1}{-x-C} - 1 \).

Step 1: Understand the Problem

We need to solve the differential equation \( \frac{dy}{dx} = (1+y)^2 \). This is a first-order differential equation, and we can use the method of separation of variables to solve it.

Step 2: Separate the Variables

To separate the variables, we'll rewrite the equation as \( \frac{dy}{(1+y)^2} = dx \). This allows us to integrate both sides separately.

Step 3: Integrate Both Sides

First, integrate the left side: \( \int \frac{dy}{(1+y)^2} \). This integral is \( -\frac{1}{1+y} + C_1 \). Now integrate the right side: \( \int dx = x + C_2 \).

Step 4: Set the Integrals Equal

Combine the results from the integrations. We'll have \( -\frac{1}{1+y} = x + C \), where \( C = C_2 - C_1 \).

Step 5: Solve for y

Rearrange the equation to solve for \( y \) in terms of \( x \). Start by isolating \( y \): Multiply both sides by \( -1 \) to get \( \frac{1}{1+y} = -x - C \). Then invert both sides: \( 1+y = \frac{1}{-x-C} \). Finally, solve for \( y \) by subtracting 1 from both sides to get \( y = \frac{1}{-x-C} - 1 \).

Step 6: Simplify the Expression

Simplify the expression for \( y \). The solution is then \( y = \frac{1 - (x + C)}{-x - C} \) or, equivalently, \( y = \frac{1}{-x-C} - 1 \).

Key Concepts

Separation of Variables

To solve certain first-order differential equations, the technique called "separation of variables" comes in handy. This method requires us to rearrange the given equation into a form where each variable appears on its respective side. Imagine you have an equation like \( \frac{dy}{dx} = (1+y)^2 \). Here, the variables need to be separated such that all terms involving \( y \) are on one side of the equation and all terms involving \( x \) are on the other side.
This is achieved by manipulating the equation to form \( \frac{dy}{(1+y)^2} = dx \).
By doing this, the equation becomes "separable" and ready for integration.

• The key in separation is to express the function in a form that allows easy integration.
• Separating the variables simplifies the solution process.
• Both sides of the equation can be handled independently and solved separately by integrating.

By ensuring that everything dependent on \( y \) and \( x \) is moved to opposite sides, separation of variables allows us to simplify the process of solving the differential equation.

First-Order Differential Equations

First-order differential equations are a type of differential equation where the highest derivative present is the first derivative. In our example, \( \frac{dy}{dx} = (1+y)^2 \), \( \frac{dy}{dx} \) is the first derivative of \( y \) with respect to \( x \).
These equations often describe simple dynamic systems and processes such as decay, growth, or motion.
To solve a first-order differential equation, you must determine a function \( y(x) \) that satisfies the equation.

• These equations may allow for special techniques like separation of variables, which often make them simpler to handle.
• First-order equations are foundational for learning about more complex differential equations.

Additionally, these equations can be linear or nonlinear. In our example, it is a nonlinear equation due to the exponent on \( (1+y) \). Identifying these characteristics helps in selecting the right method, such as separation of variables, for solving.

Integration

Integration is a mathematical operation used to solve differential equations. When variables are separated, as in \( \frac{dy}{(1+y)^2} = dx \), the next logical step leads to integrating both sides of the equation independently.
This step involves determining two antiderivatives, one on the left (\( \int \frac{dy}{(1+y)^2} \)) and another on the right (\( \int dx \)).
The integral \( \int \frac{dy}{(1+y)^2} \) results in \( -\frac{1}{1+y} + C_1 \), where \( C_1 \) is the constant of integration. Similarly, the integration of \( \int dx \) gives \( x + C_2 \), with \( C_2 \) as another constant.

• Integration transforms the differential into a solvable algebraic function.
• It helps in reverting back to the original function that describes the behavior of the system.
• Don't forget constants of integration; they are important for finding particular solutions.

Thus, integrating both sides is crucial as it allows one to find a solution represented in terms of an equation that describes all possible behaviors given the initial differential equation.