Problem 10 - Single Variable Calculus

Question

Solve the differential equation. $$ \frac{d y}{d x}+y \cot x=\cos x $$

Step-by-Step Solution

Answer

The general solution of the given differential equation is: \(y(x) = \pm\frac{1}{6}\sin^2{x} + \frac{C}{\sin x}\).

Step 1: Identify the components of the differential equation

The given differential equation is: \[ \frac{dy}{dx} + y \cot{x} = \cos{x} \] Here, the coefficient of y is \( a(x) = \cot{x}\) and the forcing term on the right-hand side is \(b(x) = \cos{x}\).

Step 2: Determine the integrating factor

To compute the integrating factor (IF), we use the formula: \[ \text{IF} = e^{\int a(x) dx} \] So for this problem, \[ \text{IF} = e^{\int \cot{x} dx} \] Now, the integral can be calculated as follows: \[ \int \cot{x} dx = \ln|\sin x| + C \] Thus, the integrating factor is: \[ \text{IF} = e^{\ln|\sin x|} = |\sin x| \]

Step 3: Multiply through by the integrating factor

Now, we multiply both sides by the integrating factor we found in the previous step: \[ |\sin x| \left( \frac{dy}{dx} + y \cot{x} \right) = |\sin x|\cos{x} \] This results in: \[ | \frac{ d( y \sin x ) }{dx} | = \sin{x}\cos{x} \]

Step 4: Integrate both sides and solve for y(x)

Next, we integrate both sides with respect to x: \[ \int | \frac{ d( y \sin x ) }{dx} | dx = \int |\sin{x}\cos{x}| dx \] On the right-hand side, we can use the substitution: \(u = \sin x\), \(du = \cos x dx\): \[ \int |u| du = \int \frac{1}{2}u^2 du = \frac{1}{6}u^3 + C \] Substituting back u = sin(x) gives: \[ \frac{1}{6}\sin^3{x} + C \] Now, on the left-hand side, we have: \[ \int | \frac{ d( y \sin x ) }{dx} | dx = | y \sin x | \] Therefore, we obtain: \[ |y \sin x| = \frac{1}{6}\sin^3{x} + C \] Dividing both sides by \(\sin x\) to solve for \(y(x)\), \[ y(x) = \pm\frac{1}{6}\sin^2{x} + \frac{C}{\sin x} \] This is the general solution of the given differential equation.

Key Concepts

Integrating Factor

The integrating factor is a key tool in solving first-order linear differential equations. It allows us to transform these equations into a more manageable form.

When given a differential equation in the form \( \frac{d y}{d x} + a(x)y = b(x) \), the integrating factor is a function derived to multiply through the equation, thereby directly making the left-hand side a derivative of a product of functions.

### Steps to Find and Use the Integrating Factor

• Identify the function \( a(x) \) as the coefficient of \( y \) in the differential equation.
• Calculate the integrating factor using the formula: \( \text{IF} = e^{\int a(x) dx} \).
• Multiply the entire differential equation by this integrating factor to simplify it.
• This manipulation allows the left-hand side to become a straightforward derivative we can integrate easily.

In the example, \( a(x) = \cot{x} \), hence the integrating factor results in \( \text{IF} = |\sin x| \), transforming the equation to allow simple integration.

Separable Differential Equations

Separable differential equations are a special class where the equation can be split into two separate integrals. This makes it easier to solve as each variable can be worked separately.

In cases where two functions of different variables (one with \(x\) and another with \(y\)) are multiplicatively related, we can use separable equations technique.

### Key Steps in Solving Separable Equations

• Rewrite the equation so each side of the equation pertains to a different variable.
• Integrate both sides independently.
• Solve the integrals to find the relationship between \(x\) and \(y\).

Although the example is a linear differential equation, understanding separable techniques enriches problem-solving skills as it establishes a foundation of moving between different differential forms.

First Order Linear Differential Equations

First-order linear differential equations form the basis of many concepts in differential equations. These equations are of the form \( \frac{dy}{dx} + a(x)y = b(x) \), which are linear in terms of \( y \).

These are called 'first order' because they involve the first derivative of a function without going any higher.

### Solving First Order Linear Differential Equations

• Identify the structure: isolate the derivative and coefficient of \( y \).
• Use the integrating factor method to simplify the differential equation.
• Solve the resulting simplified, exact equation by integrating both sides.

In the example provided, identifying the linear format of the equation allowed the use of an integrating factor, simplifying it to \( |\frac{d(y \sin x)}{dx}| \), thus making it straightforward to solve through integration.