Problem 11 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(11-20,\) solve the initial value problem explicitly. $$\frac{d y}{d x}=3 \sin x\( and \)y=2\( when \)x=0$$

Step-by-Step Solution

Answer

The solution to the initial value problem is \(y = -3 \cos x + 5\).

Step 1: Integration

Start by integrating the right side of the given differential equation. As the integral of \(3 \sin x\) is \(-3 \cos x\), the result is \(y= -3 \cos x + C\), where \(C\) is the constant of integration.

Step 2: Compute the Constant

Use the initial value \(y=2\) when \(x=0\) to compute the constant. Such that, when we plug \(x = 0\) into the function, we get \(y = -3 \cos 0 + C\). Solving this equation will give \(C = 2 + 3 = 5\).

Step 3: Derive the Final Solution

Substituting \(C = 5\) into \(y = -3 \cos x + C\) gives the final explicit solution to the initial value problem, which will be \(y = -3 \cos x + 5\).

Key Concepts

Differential Equation

A differential equation is a mathematical equation that relates some function with its derivatives. In practice, these types of equations are powerful tools for describing relationships where a rate of change or a particular dynamic is observed. In your homework problem, the differential equation given is \frac{d y}{d x}=3 \text{sin} x\. This represents the rate of change of a function, \( y \), with respect to another function, \( x \), which is a common type of first-order differential equation. Solving a differential equation involves finding the function \( y \) that satisfies the given relationship for its derivative.

In the context of an initial value problem, you're given an extra piece of information: the value of \( y \) when \( x \) equals a specific number. In your example, the problem provides the initial condition \( y=2 \) when \( x=0 \), which allows you to solve the differential equation uniquely.

Integration

Integration is the process of finding the integral of a function, which essentially represents the accumulation of quantities, such as areas under curves. When you integrate the differential equation, you transition from the rate of change (the derivative) to the quantity itself (the function). Integrating the given function \( 3 \text{sin} x \), you find the related function whose derivative would generate \( 3 \text{sin} x \).The integral of \( 3 \text{sin} x \) is \( -3 \text{cos} x + C \), where \( C \) represents the constant of integration. Integration is crucial in solving differential equations, as it lets you move from knowing how a function is changing, to understanding the function's overall behavior.

Constant of Integration

The constant of integration, denoted as \( C \), emerges every time you integrate an indefinite integral. Since the process of differentiation of a constant is zero, when you integrate, there are infinitely many solutions that differ by a constant. This is why it's called an 'indefinite' integral. To identify the correct constant of integration in an initial value problem, you use the provided initial conditions.In the solution to your exercise, the initial condition was used: \( y = 2 \) when \( x = 0 \). By substituting these values into the integrated function, you solve for \( C \) and find that it equals 5. Therefore, your final solution includes this constant: \( y = -3 \text{cos} x + 5 \), making it a definite, or specific, solution to the original differential equation.