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Chapter 3: Problem 65

Solve and graph solutions of the differential equation \((d y / d x)=2 x\)

Short Answer

Expert verified
The general solution of the differential equation \(\frac{dy}{dx} = 2x\) is \(y = x^2 + C\), where C is the constant of integration. The graph of the general solution represents a family of parabolas with different y-intercepts, opening upwards. Different values of C will shift the parabola along the y-axis.

Step by step solution

01

Integrate the differential equation

The given differential equation is \(\frac{dy}{dx} = 2x\). First, rewrite the differential equation using the separation of variables technique: \(dy = 2x dx\) Now integrate both sides with respect to their respective variables: \[\int dy = \int 2x dx\] Upon integration, we have: \(y = x^2 + C\), where C is the constant of integration. This is the general solution of our differential equation.
02

Graph the general solution

Now that we have our general equation, let's graph it. The equation \(y = x^2 + C\) represents a family of parabolas with different y-intercepts, all with the same basic shape of a parabola opening upwards. 1. If C = 0, we have the base function y = x^2, which passes through the origin. 2. If C > 0, the parabola shifts upward along the y-axis. 3. If C < 0, the parabola shifts downward along the y-axis. To visualize this, you can graph several instances of the general solution for different values of C, such as: - \(y = x^2 + 1\) (C = 1, parabola shifted 1 unit up) - \(y = x^2\) (C = 0, base function) - \(y = x^2 - 1\) (C = -1, parabola shifted 1 unit down) It's important to note that these graphs are merely examples of specific solutions from the general solution. The general solution includes all possible parabolas with different y-intercepts depending on the value of the constant C.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Separation of Variables
Separation of variables is a common method used to solve differential equations, which involves rearranging the equation to isolate one variable and its differential on one side, and the other variable and its differential on the other side. This allows us to integrate each side independently.

For example, in our exercise, the differential equation \(\frac{dy}{dx} = 2x\) can be separated by moving all terms containing \(y\) to one side and those containing \(x\) to the other, resulting in \(dy = 2x dx\). This step prepares us for integrating both sides with respect to their own variables. Through this process, we're able to find a general solution to the equation.
Integration of Functions
Integration is a fundamental concept in calculus that refers to the process of finding the integral of a function, which is essentially the 'opposite' of taking the derivative. It deals with the accumulation of quantities, such as areas under curves. When we integrate a function, we're finding the antiderivative or the original function before differentiation.

Our exercise necessitated integrating both sides of the equation where one side had \(dy\), which integrates to \(y\), and the other side had \(2x dx\), which integrates to \(x^2\). When we integrate, we must not forget to add the constant of integration (\(C\)) because there are infinitely many antiderivatives for a given function, each varying by a constant.
Graphing Parabolas
A parabola is a specific type of graph produced by a quadratic function of the form \(y = ax^2+ bx +c\), where \(a\), \(b\), and \(c\) are constants. Parabolas can open upwards or downwards, depending on the sign of \(a\), and can be shifted horizontally and vertically by adjusting \(b\) and \(c\).

In our context, after integrating the given differential equation, the result is a quadratic equation representing a family of parabolas all opening upwards (since the coefficient of \(x^2\) is positive). The graph varies only in its vertical position based on the constant \(C\). Understanding how to graph these equations is crucial as it enables us to visualize the solutions to our differential equation.
Constant of Integration
The constant of integration is an essential part of the integration process, especially when dealing with indefinite integrals. It represents an unknown constant that we add to the antiderivative because the process of differentiation of a constant results in zero. The presence of this constant signifies that there is an infinite number of functions that could be the original antiderivative.

In our exercise, when we integrated \(2x dx\) to get \(x^2\), we had to add a constant \(C\). This constant affects the y-intercept of the graphed parabola allowing for an entire family of solutions to the differential equation, each corresponding to a different initial condition that could provide a specific value for \(C\). The constant of integration is crucial; without it, the solution would be incomplete.

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Most popular questions from this chapter

What technique should be used for solving the differential equation $$ \left(x+y^{2}\right) d y+\left(y-x^{2}\right) d x=0 ? $$

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