Question

Finding a General Solution Using Separation of Variables In Exercises \(1-14,\) find the general solution of the differential equation. $$ y y^{\prime}=4 \sin x $$

Short answer

The general solution of the given differential equation is \[y = \sqrt{-2\cos(x)+2C}.\]

Step-by-step solution

Separate Variables

Start by separating the variables \(y\) and \(x\) on each side of the equation: \[\frac{y}{4 \sin x} dy = dx \]

Integrate Both Sides

Once the equation is separated into parts including \(x\) and \(y\), every side can be integrated with respect to its own variable: \[\int\frac{y}{4 \sin x} dy = \int dx.\] After integrating, it becomes: \[-\frac{1}{4}\cos(x) = \frac{1}{2}y^2 + C \] where \(C\) is the constant of integration.

Rearrange the Equation

After the integration, rearrange the equation to express it in terms of \(y\): \[y = \sqrt{-2\cos(x)+2C}.\]

Key concepts

Differential Equations

Differential equations are mathematical equations that involve rates at which quantities change. They are a fundamental tool in describing various physical and theoretical processes within mathematics, engineering, and science. These equations relate functions of one or more variables with their derivatives.

In the exercise provided, we encounter a first-order differential equation where the rate of change of a function, denoted by the derivative \(y'\), is proportional to a trigonometric function of another variable \(x\). Solving differential equations often involves finding a function that satisfies the equation, known as the equation's solution. In this case, the goal is to find the general solution for \(y\) in terms of \(x\).

The process of separation of variables is one method of tackling such equations. It relies on rearranging the equation so that each variable and its differential are on separate sides of the equation, allowing integration to be performed with respect to the individual variables.

Integrating Factors

Integrating factors are a tool used to solve certain types of differential equations, particularly when the separation of variables is not possible. An integrating factor is a function that, when multiplied with a given differential equation, transforms it into an equivalent equation that is easier to solve, typically because it becomes an exact equation.

While the exercise provided does not require the use of integrating factors, it is useful to understand this concept for more complex equations where separation of variables is not an option. The process typically involves finding a function of one of the variables that, when used to multiply each term in the differential equation, yields an expression that allows the left- and right-hand sides to be represented as the derivative of a product of functions, which can then be integrated easily. The integrating factor method is especially powerful for linear differential equations of the first order.

General Solution of Differential Equations

The general solution of a differential equation represents the set of all possible solutions that satisfy the equation, and it typically includes a constant of integration denoted by \(C\). This constant represents an infinite number of solutions, as it can take on any value. In essence, the general solution accounts for the initial conditions that are not specified within the problem.

In our exercise, the general solution to the first-order differential equation is found to be \(y = \sqrt{-2\cos(x) + 2C}\). This equation represents the infinite family of curves that satisfy the original differential equation for any particular value of \(C\). It's important to note that specifying a particular initial condition would lead us to a specific solution, which is a single curve from this family that passes through the point defined by the initial condition.