Question

For the following problems, find the general solution to the differential equation.$y^{\prime }=y$

Short answer

The general solution is $y=C{e}^x$.

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is $y^{\prime }=y$. This is a first-order linear ordinary differential equation.

Separate Variables

Rewrite the differential equation to separate variables. Divide both sides by $y$ to get $\frac{dy}{y}=dx$.

Integrate Both Sides

Integrate $\frac{dy}{y}$ with respect to $y$ and $dx$ with respect to $x$. This gives us $\int \frac{1}{y}dy=\int 1dx$, resulting in $\ln |y|=x+C$, where $C$ is the integration constant.

Solve for $y$

Exponentiate both sides to solve for $y$. We have $e^{\ln |y|}=e^{x+C}$, which simplifies to $|y|=e^x\cdot e^C$. Let $e^C=C_1$, a new constant, so $y=C_1{e}^x$.

Determine the General Solution

The general solution to the differential equation is $y=C{e}^x$, where $C$ is any real constant.

Key concepts

First-Order Differential Equations

First-order differential equations are a fundamental tool in mathematics. They describe relationships where the rate of change of a variable is directly linked to another variable. In simple terms, these equations involve derivatives of a function with respect to only one variable, usually denoted as $y^{\prime }$ or $\frac{dy}{dx}$.

First-order simply refers to the highest derivative in the equation being the first derivative. A linear first-order differential equation takes the form $y^{\prime }+P(x)y=Q(x)$. Here, $P(x)$ and $Q(x)$ are functions of $x$, or constants in some cases.

These equations appear in various real-life scenarios:

• Population growth models, where growth rate depends on current population size.
• Radioactive decay, where the decay rate is proportional to the amount present.
• Cooling of objects, where the rate of heat loss is proportional to the temperature difference from its surroundings.

Understanding first-order differential equations helps us model, analyze, and predict behaviors in different fields, from science to engineering.

Separation of Variables

Separation of variables is a method designed to solve first-order differential equations. The goal is to isolate the functions of two different variables on opposite sides of the equation, generally represented as $y$ and $x$. This method is applicable when the differential equation can be expressed as a product of functions like $\frac{dy}{dx}=g(y)h(x)$.

To execute separation of variables, follow these steps:

• Step 1: Rearrange the equation so that all terms containing $y$ are on one side and $x$ terms on the other. For example, $\frac{dy}{y}=dx$.
• Step 2: Integrate both sides. This helps in finding the implicit solution to the differential equation.

This method works well when the two sets of variables don't mix, simplifying the integration process. It's simple and efficient where applicable, making it a popular choice for solving specific types of differential equations.

Integration

Integration is a vital concept in solving differential equations. It is the formal way of finding the antiderivative or the original function before differentiation. In the context of separation of variables, integration allows us to find the solution from the differentiated form.

When we solve $\frac{dy}{y}=dx$, we apply integration to both sides:

• The left side, $\int \frac{1}{y}dy$, simplifies to $\ln |y|$.
• The right side, $\int 1dx$, simplifies to $x$.

Combining these results gives us $\ln |y|=x+C$, where $C$ is an integration constant. Integration constants are crucial as they account for all possible solutions by accounting for the indefinite behaviors inherent in antiple integrations.

Thus, the integration step provides a general solution to the differential equation. In many practical problems, further steps, like exponentiation, are needed to solve explicitly for the variables involved, yielding a comprehensible form of the solution. By integrating, we transition from a rate of change back to the original function, allowing us to explore the behavior of the system represented by the differential equation.