Question

Solving a Differential Equation In Exercises \(1-10\) , solve the differential equation. $$ x y+y^{\prime}=100 x $$

Short answer

The solution to the differential equation \(xy+y'=100x\) is \(y = 50x^2 - 0.5x^2y + C\). C is the constant of integration.

Step-by-step solution

Rearrange the Equation

Rearrange the given equation by isolating \(y'\) on one side of the equation. After this step, you will have: \(y' = 100x - xy\)

Integrate Both Sides

Once we have the equation in this form, we can integrate both sides of the equation with respect to x. This will result in the equation: \(\int y' dx = \int (100x - xy) dx\)

Solve the Integration

Integration of both sides yields an unknown function C (constant of integration) which we can't determine with the given information, resulting in the equation: \(y = 50x^2 - 0.5x^2y + C\)

Key concepts

Integration

Integration is a fundamental concept in calculus, mainly used to find areas under curves, among other applications. In the context of solving differential equations, integration helps us find a function when its derivative is given. To integrate means to accumulate values, which sometimes can be visualized as finding the total area under a graph of a function.
There are several methods of integration, each suited for different kinds of problems:

• Definite Integration: This computes the accumulation of quantities over an interval, resulting in a specific number.
• Indefinite Integration: This determines a general form of the antiderivative, including an arbitrary constant, often denoted as 'C'.
• Substitution Method: Useful for integrals that involve complex expressions, by simplifying them into more manageable pieces.
• Integration by Parts: Breaks a complex integral into simpler parts to solve.

In differential equations, like the one given in the exercise, integration is used to combine both sides of an equation to simplify and find a solution that satisfies the original differential equation.

Constant of Integration

The constant of integration, symbolized as 'C', plays a critical role in the integration process, particularly when dealing with indefinite integrals. When we integrate a function, the result includes this constant because several functions can share the same derivative. Essentially, the constant represents an infinite number of possible solutions that fit the differential equation.

Whenever you integrate without specific boundary conditions, the constant remains unspecified. Adding this constant acknowledges that:

• Many forms of a function differ only by a constant yet share the same derivative.
• It is crucial for finding the particular solution, which matches additional conditions or constraints.

In the example provided, the result of the integration included the constant 'C'. This is because we lacked initial conditions to pinpoint a singular solution. Therefore, the equation had to account for this uncertainty, accommodating multiple potential solutions.

Differential Calculus

Differential calculus focuses on the concept of the derivative, which represents the rate at which a function is changing. This branch of mathematics is concerned with how something changes instantaneously, identifying slopes of curves, and finding the rate of change over intervals. In solving differential equations, the objective is often to determine the unknown function that is related to its derivative, giving insights into the dynamics and behavior of systems.

In practical terms, differential calculus helps to:

• Analyze how functions behave.
• Find maxima and minima points, which are critical in optimization problems.
• Model scenarios in physics, economics, and other sciences where change is a key factor.

For the exercise given, differential calculus provides the framework to solve for the unknown function 'y' by connecting it to its rate of change 'y"]'). By expressing the rate of change in a manageable form, we could then use integration to backtrack into the original function, solving the differential equation provided.