Given The Differential Equation $\frac{d Y}{d X} = \frac{x+5}{2 Y}$, Find The Particular Solution, $y=f(x$\], With The Initial Condition $f(6) = -5$.

Separable Differential Equations: A Step-by-Step Guide to Solving $\frac{d y}{d x} = \frac{x+5}{2 y}$

In this article, we will delve into the world of separable differential equations and explore the solution to the given differential equation $\frac{d y}{d x} = \frac{x+5}{2 y}$. We will use the method of separation of variables to find the particular solution, $y=f(x)$, with the initial condition $f(6) = -5$.

What are Separable Differential Equations?

A separable differential equation is a type of differential equation that can be written in the form $\frac{d y}{d x} = f(x)g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$. The key characteristic of separable differential equations is that they can be separated into two distinct functions, one depending on $x$ and the other on $y$. This allows us to solve the equation by integrating both sides separately.

The Method of Separation of Variables

To solve the given differential equation, we will use the method of separation of variables. This involves separating the variables $x$ and $y$ and then integrating both sides separately. The steps involved in this method are as follows:

1. Separate the variables: We start by separating the variables $x$ and $y$ by multiplying both sides of the equation by $2y$ and dividing both sides by $x+5$. This gives us the equation $\frac{d y}{d x} \cdot 2y = x+5$.
2. Integrate both sides: We then integrate both sides of the equation with respect to $x$. This gives us the equation $2y \cdot \frac{d y}{d x} = \int (x+5) dx$.
3. Solve for y: We then solve for $y$ by integrating both sides of the equation with respect to $x$. This gives us the equation $y^2 = \int (x+5) dx + C$, where $C$ is the constant of integration.

Solving the Differential Equation

Now that we have separated the variables and integrated both sides, we can solve for $y$. We start by evaluating the integral on the right-hand side of the equation. This gives us the equation $y^2 = \frac{x^2}{2} + 5x + C$.

Applying the Initial Condition

We are given the initial condition $f(6) = -5$. We can use this condition to find the value of the constant $C$. We substitute $x=6$ and $y=-5$ into the equation $y^2 = \frac{x^2}{2} + 5x + C$. This gives us the equation $(-5)^2 = \frac{6^2}{2} + 5(6) + C$.

Simplifying the Equation

We can simplify the equation by evaluating the expressions on the right-hand side. This gives us the equation $25 = 18 + 30 + C$.

Solving for C

We can solve for $C$ by subtracting $18 + 30$ from both sides of the equation. This gives us the equation $C = -23$.

Finding the Particular Solution

Now that we have found the value of the constant $C$, we can find the particular solution, $y=f(x)$. We substitute $C=-23$ into the equation $y^2 = \frac{x^2}{2} + 5x - 23$. This gives us the equation $y^2 = \frac{x^2}{2} + 5x - 23$.

Solving for y

We can solve for $y$ by taking the square root of both sides of the equation. This gives us the equation $y = \pm \sqrt{\frac{x^2}{2} + 5x - 23}$.

In this article, we have used the method of separation of variables to solve the differential equation $\frac{d y}{d x} = \frac{x+5}{2 y}$. We have found the particular solution, $y=f(x)$, with the initial condition $f(6) = -5$. The solution is given by the equation $y = \pm \sqrt{\frac{x^2}{2} + 5x - 23}$.

The final answer is $\boxed{y = \pm \sqrt{\frac{x^2}{2} + 5x - 23}}$.
Separable Differential Equations: A Q&A Guide

In our previous article, we explored the solution to the differential equation $\frac{d y}{d x} = \frac{x+5}{2 y}$ using the method of separation of variables. In this article, we will provide a Q&A guide to help you better understand the concept of separable differential equations and how to solve them.

Q: What is a separable differential equation?

A: A separable differential equation is a type of differential equation that can be written in the form $\frac{d y}{d x} = f(x)g(y)$, where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$.

Q: How do I know if a differential equation is separable?

A: To determine if a differential equation is separable, you need to check if it can be written in the form $\frac{d y}{d x} = f(x)g(y)$. If it can be written in this form, then it is a separable differential equation.

Q: What is the method of separation of variables?

A: The method of separation of variables is a technique used to solve separable differential equations. It involves separating the variables $x$ and $y$ and then integrating both sides separately.

Q: How do I apply the method of separation of variables?

A: To apply the method of separation of variables, you need to follow these steps:

1. Separate the variables: Multiply both sides of the equation by $2y$ and divide both sides by $x+5$.
2. Integrate both sides: Integrate both sides of the equation with respect to $x$.
3. Solve for y: Solve for $y$ by integrating both sides of the equation with respect to $x$.

Q: What is the role of the constant of integration in separable differential equations?

A: The constant of integration is a constant that is added to the solution of a differential equation to make it complete. In separable differential equations, the constant of integration is used to account for the arbitrary constant that is introduced when integrating both sides of the equation.

Q: How do I apply the initial condition to a separable differential equation?

A: To apply the initial condition to a separable differential equation, you need to substitute the initial condition into the solution of the equation and solve for the constant of integration.

Q: What is the significance of the particular solution in separable differential equations?

A: The particular solution is a specific solution to a differential equation that satisfies the initial condition. In separable differential equations, the particular solution is used to find the value of the constant of integration.

Q: Can I use the method of separation of variables to solve non-separable differential equations?

A: No, the method of separation of variables can only be used to solve separable differential equations. If a differential equation is not separable, then you need to use a different method to solve it.

In this article, we have provided a Q&A guide to help you better understand the concept of separable differential equations and how to solve them. We hope that this guide has been helpful in clarifying any doubts you may have had about separable differential equations.

The final answer is $\boxed{y = \pm \sqrt{\frac{x^2}{2} + 5x - 23}}$.