Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Solve the given differential equation. \(\frac{d y}{d x}=x y^{2}\)

Step-by-Step Solution

Answer

The solution to the differential equation is \( y = \frac{-1}{2x^{2}+C}\).

Step 1: Separate the variables

Write the differential equation in the form that will allow us to separate the variables. We can do this by dividing both sides of the equation by \(y^{2}\) and multiplying both sides by \(dx\), resulting in: \(\frac{1}{y^{2}}\frac{dy}{dx} = x\).

Step 2: Integrate both sides

Now we can easily integrate both sides of the equation. The left side, \(\int_{ }^{} \frac{1}{y^{2}} dy\), turns to \(-\frac{1}{y}\). The right side, \(\int_{ }^{} x dx\), turns to \(\frac{1}{2}x^{2}\). So, we get: \(-\frac{1}{y} = \frac{1}{2}x^{2} + C\), where C is the constant of integration.

Step 3: Solve for y

Finally, we solve for y by multiplying by -1 to both sides, which gives: \( \frac{1}{y} = -\frac{1}{2}x^{2} - C \). The inverse operation reveals the solution to be: \( y = \frac{-1}{2x^{2}+C}\).

Key Concepts

Separation of Variables

When faced with a differential equation like \( \frac{d y}{d x}=x y^{2} \), one effective method to find a solution is using the technique known as separation of variables. This is a strategy often employed to solve first-order ordinary differential equations, which involves rearranging the equation to isolate the differential components of each variable on opposite sides.

For instance, in our equation, we can divide through by \( y^{2} \) to separate y from x, creating two integrable sides: one involving only y (\( \frac{1}{y^{2}} \)) and another involving only x (x). The crucial idea here is to transform the original equation into a form where all terms including y (and dy) are on one side, and all terms including x (and dx) are on the other, setting the stage for integration, which is the next pivotal step in finding the solution to the differential equation.

Integration Techniques

Integration is at the heart of solving differential equations, with various techniques available depending on the form of the equation. In the example \( \frac{d y}{d x}=x y^{2} \), once the variables are separated, we encounter integrals that we need to evaluate: \( \int \frac{1}{y^{2}} dy \) and \( \int x dx \).

For the first integral, we recognize a common formula: \( \int u^{-n} du = \frac{u^{-(n-1)}}{-(n-1)} + C \) for \( n eq 1\), which we apply to find \( -\frac{1}{y} \). The second integral is straightforward power rule integration, yielding \( \frac{1}{2}x^{2} \). This step underscores the importance of being familiar with basic integration rules and techniques such as u-substitution, integration by parts, and recognizing common integral forms. Developing these skills will greatly enhance a student’s ability to solve a wide range of differential equations.

Differential Equation Solutions

After integrating both sides of a separated equation, the next step is to explicitly solve for the function we are interested in—in this case, y, the dependent variable. We arrived at \( -\frac{1}{y} = \frac{1}{2}x^{2} + C \) through separation of variables and integration; this implicitly defines y in terms of x and the integration constant C.

By manipulating this expression algebraically, we can find y explicitly. Multiplying both sides by -1 and then taking the reciprocal of both sides, we derive \( y = \frac{-1}{2x^{2}+C} \). The constant C represents an infinite number of solutions corresponding to a family of curves on the xy-plane. Each curve represents a specific solution to the original differential equation, with the constant C determined by initial conditions, if they are given. This illustrates the fundamental concept that solving a differential equation often results in a general solution inclusive of an arbitrary constant representing a suite of possible specific solutions.