Question

Find the solution to the initial value problem. $$ y^{\prime}=3 x-\cos x+2, y(0)=4 $$

Short answer

The solution is \( y = \frac{3}{2}x^2 - \sin x + 2x + 4 \).

Step-by-step solution

Identify the Differential Equation and Initial Condition

The given problem is a first-order ordinary differential equation (ODE): \( y' = 3x - \cos x + 2 \). The initial condition provided is \( y(0) = 4 \). Our task is to find the function \( y(x) \) that satisfies both the differential equation and the initial condition.

Integrate the Differential Equation

To solve for \( y \), we integrate the right-hand side of the differential equation with respect to \( x \). The integral to evaluate is \( \int (3x - \cos x + 2) \ dx \). Break this into parts: \( \int 3x \ dx \), \( \int -\cos x \ dx \), and \( \int 2 \ dx \).

Calculate Each Integral Separately

Integrate each term:- \( \int 3x \ dx = \frac{3}{2}x^2 \) (using the power rule)- \( \int -\cos x \ dx = -\sin x \) (since the derivative of \( \sin x \) is \( \cos x \))- \( \int 2 \ dx = 2x \) (constant integration rule).Combine these results to find the integral of the whole expression.

Combine the Results and Add Constant of Integration

Summing the results, we get: \[ y = \frac{3}{2}x^2 - \sin x + 2x + C \]. Here, \( C \) is the constant of integration that we'll determine using the initial condition.

Apply the Initial Condition

Use the initial condition \( y(0) = 4 \) to solve for \( C \). Substitute \( x = 0 \) into the integrated function: \( 4 = \frac{3}{2}(0)^2 - \sin(0) + 2(0) + C \). This simplifies to \( 4 = C \). So, \( C = 4 \).

Write the Final Solution

Substitute \( C = 4 \) back into the integrated expression to get the final solution of the ODE: \[ y = \frac{3}{2}x^2 - \sin x + 2x + 4 \]. This function satisfies both the differential equation and the initial condition.

Key concepts

Ordinary Differential Equation

An ordinary differential equation (ODE) is an equation that involves a function and its derivatives. In our context, the goal is to find the unknown function. The ODE provided in our problem is of the form \( y' = 3x - \cos x + 2 \), which is a first-order differential equation. A first-order ODE means it involves the first derivative of the function. The notation \( y' \) represents the derivative of \( y \) with respect to \( x \). Solving such equations often involves finding a function, \( y(x) \), that satisfies the relationship defined by the ODE over some interval.

Integration

Integration is the mathematical process used to solve differential equations like the one given in the problem. When we integrate the function, we essentially find a function that originally had a derivative of the expression we integrated.
The expression \( \int (3x - \cos x + 2) \, dx \) needs to be solved. Integration is performed term by term here, producing three separate integrals, which simplifies analysis:

• The integral \( \int 3x \, dx = \frac{3}{2}x^2 \) is found using the power rule.
• The integral \( \int -\cos x \, dx = -\sin x \) uses knowledge of basic trigonometric integration.
• Finally, \( \int 2 \, dx = 2x \) is a straightforward integration of a constant.

Combining these results gives us the integrated function which includes a constant, \( C \), due to the indefinite nature of the integration process.

Initial Condition

The initial condition in a differential equation provides specific values for the function that help determine a unique solution. In our task, the initial condition is \( y(0) = 4 \).
This condition means that when \( x = 0 \), the value of \( y \) is 4. By applying this to the integrated equation \( y = \frac{3}{2}x^2 - \sin x + 2x + C \), we substitute \( x = 0 \) and \( y = 4 \). As a result, we find the value of the constant \( C \) in the solution which is critical for tailoring the function to meet the initial specification.

Constant of Integration

The constant of integration, denoted as \( C \), appears when solving an indefinite integral. It represents all the possible constants that could have been differentiated to produce the integrand.
In our solution, once we integrate \( 3x - \cos x + 2 \), we obtain \( y = \frac{3}{2}x^2 - \sin x + 2x + C \). Without initial conditions, \( C \) could potentially be any real number.
The initial condition \( y(0) = 4 \) helps us calculate this constant precisely by substituting into this solved equation. We determined:\( 4 = C \). Once \( C \) is known, the specific solution that satisfies the initial value problem is fully defined, and we arrive at the final solution: \( y = \frac{3}{2}x^2 - \sin x + 2x + 4 \).