Question

Finding a General Solution In Exercises \(41-52,\) use integration to find a general solution of the differential equation. $$ \frac{d y}{d x}=10 x^{4}-2 x^{3} $$

Short answer

The solution to the differential equation is y = \(2x^5 - 0.5x^4 + C\), where C is an arbitrary constant.

Step-by-step solution

Identify the Form

Identify the form of the differential equation. The given differential equation is of the form dy/dx = f(x). In this case, f(x) = 10x^4 - 2x^3. This form of differential equations can be solved by integrating both sides.

Integrate Both Sides

As the next step, integrate both sides of the equation with respect to x. So, the integral of dy/dx becomes the integral of y with respect to x on the left side, and the integral of f(x) dx on the right side. That is, we need to compute \[ \int y dx = \int (10x^4 - 2x^3) dx \].

Perform the Integral

Perform the integral on both sides. On the left side, the integral of y dx is y itself. On the right side, the integral of 10x^4 dx is 10/5*x^5 = 2x^5, and the integral of -2x^3 dx is -2/4*x^4 = -0.5x^4. Adding the two integrals, we get y = 2x^5 - 0.5x^4 + C, where C is the constant of integration.

Key concepts

Integration Techniques

Integration is the process of finding the antiderivative of a function, which is essential in solving differential equations. A differential equation relates a function with its derivatives, and often the solution involves finding an original function before it was differentiated, which is where integration comes in. There are several integration techniques, including power-rule integration, substitution, integration by parts, partial fractions, and trigonometric substitution.

For the given exercise, where we need to integrate polynomials, the power rule is the most straightforward technique. This rule states that the integral of a monomial, such as x^n, is given by x^(n+1)/(n+1), as long as n is not equal to -1. This technique was used in solving the given problem where the powers of x were simply incremented by one and then divided by the new exponent.

General Solution of Differential Equation

The general solution to a differential equation represents a family of solutions containing arbitrary constants, as opposed to a particular solution which is fixed by initial conditions. The general solution takes into account all possible antiderivatives of the function.

When we integrate a function to solve a differential equation, what we're effectively doing is finding this general solution. For example, in our exercise, once we integrate the function 10x^4 - 2x^3, we are finding all possible functions y that when differentiated, would give us back the original function of x. This is why the integration process is so critical for solving differential equations—it provides us with the broader solution that we can then refine if we have more information or initial conditions.

Constant of Integration

When we integrate a function, we add a constant of integration, typically denoted as C, to the solution. This constant represents all possible vertical shifts of the antiderivative and is a crucial component of the general solution to a differential equation.

Different values for C correspond to different members of a family of curves, each of which satisfies the differential equation. The reason behind this is because differentiation wipes out constants—the derivative of a constant is zero, so when we reverse the process through integration, we have to include this constant to account for all possible original functions. In our exercise example, adding C to the integrated polynomial 2x^5 - 0.5x^4 ensures we have the complete set of solutions.

Integrating Polynomials

Polynomials are expressions that consist of variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Integrating polynomials is a key skill in calculus, often used in solving differential equations.

The power rule for integration, as mentioned earlier, is typically used for integrating polynomials. Simply put, when we integrate a polynomial term by term, we apply the power rule to each term of the polynomial separately and then combine the results. It's important to remember to add the constant of integration at the end. In the example provided, we integrated a polynomial of the form 10x^4 - 2x^3 by applying the power rule to each term, resulting in 2x^5 - 0.5x^4 + C as our general solution.