Elimination of arbitrary constant of sin(dy/dx)=C

To eliminate the arbitrary constant from the equation \(\sin\left(\frac{dy}{dx}\right) = C\), we will follow a step-by-step process:

### Step 1: Solve for \(\frac{dy}{dx}\)
The given equation is:

\[
\sin\left(\frac{dy}{dx}\right) = C
\]

Take the inverse sine (arcsin) on both sides:

\[
\frac{dy}{dx} = \sin^{-1}(C)
\]

Let \(\alpha = \sin^{-1}(C)\), which is a constant because \(C\) is constant. So,

\[
\frac{dy}{dx} = \alpha
\]

### Step 2: Integrate both sides
Now, integrate both sides with respect to \(x\):

\[
y = \alpha x + D
\]

where \(D\) is the constant of integration.

### Step 3: Eliminate arbitrary constants
Now, we need to eliminate the arbitrary constant \(D\). To eliminate it, differentiate \(y = \alpha x + D\) with respect to \(x\):

\[
\frac{dy}{dx} = \alpha
\]

Thus, after differentiating, the arbitrary constant \(D\) is eliminated.

Hence, the solution to the differential equation with the arbitrary constant eliminated is:

\[
\frac{dy}{dx} = \sin^{-1}(C)
\]

### Summary
By solving and differentiating, we find that the equation \(\sin\left(\frac{dy}{dx}\right) = C\) reduces to \(\frac{dy}{dx} = \sin^{-1}(C)\), eliminating any arbitrary constant from the equation.

Do you want further clarification on any of these steps or more details?

---

Here are five questions related to the topic:

1. What is the process of solving a differential equation with an arbitrary constant?
2. How can inverse trigonometric functions help in solving differential equations?
3. What are the implications of the constant of integration in such problems?
4. How does one verify the solution to a first-order differential equation?
5. What are other methods to eliminate arbitrary constants in differential equations?

**Tip:** Always check for valid ranges when dealing with inverse trigonometric functions, as they have restricted domains and ranges.

Elimination of arbitrary constant of sin(dy/dx) = C

Learn how to eliminate the arbitrary constant in the differential equation sin(dy/dx) = C with a detailed step-by-step solution. This problem covers inverse trigonometric functions, integration, and solving first-order differential equations. Suitable for undergraduate students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation