Question

Solve the given differential equations. $$ \frac{d y}{d x}+\frac{3 y}{x}=6 x^{2} $$

Short answer

The general solution for the given differential equation \(\frac{dy}{dx} + \frac{3y}{x} = 6x^2\) is \(y(x) = x^3 + \frac{C}{x^3}\), where C is the constant of integration.

Step-by-step solution

Identify the form of the given differential equation

The given differential equation is in the form: $$ \frac{dy}{dx} + P(x)y = Q(x) $$ In our case, $$P(x) = \frac{3}{x}$$ and $$Q(x) = 6x^2$$.

Determine the Integrating Factor

The integrating factor (IF) is given by the formula: $$ IF = e^{\int P(x) dx} $$ For our problem, we have: $$ IF = e^{\int \frac{3}{x} dx} $$ Now we integrate \(P(x)\) to find the IF: $$ IF = e^{3 \ln{x}} = x^{3} $$

Multiply the whole equation by the Integrating Factor

Next, we multiply the entire differential equation by the Integrating Factor (IF): $$ x^3\frac{dy}{dx} + 3x^2y = 6x^5 $$

Integrate both sides of the resulting equation

Now, integrate both sides of the equation with respect to x: $$ \int \left( x^3\frac{dy}{dx} + 3x^2y \right) dx= \int 6x^5 dx $$ Notice that the left side of the equation is the derivative of the product of the integrating factor and y(x): $$ \frac{d}{dx}\left( x^3y \right) = 6x^5 $$ Now, we integrate both sides: $$ \int \frac{d}{dx}\left( x^3y \right) dx = \int 6x^5 dx $$ This gives us: $$ x^3y = x^6 + C $$

Solve for the function y(x)

Finally, we solve for the function y(x): $$ y(x) = \frac{x^6 + C}{x^3} $$ So, the general solution for the given differential equation is: $$ y(x) = x^3 + \frac{C}{x^3} $$

Key concepts

Integrating Factor

When solving first-order linear differential equations, an integrating factor is a function that we multiply by the equation to facilitate integration. It's a pivotal concept that transforms a differential equation into an exact one, making integration manageable.

The integrating factor is calculated using the formula:
\[ IF = e^{\int P(x) dx} \] Where P(x) is a function of x from the standard form of the differential equation (dy/dx) + P(x)y = Q(x). It's important to note that the integrating factor relies purely on the function P(x), and its purpose is to create a situation where the left side of the differential equation can be expressed as the derivative of a product.

For the given exercise:
\[ IF = e^{3 \ln{x}} = x^{3} \] The integrating factor here is x^3. When you multiply the entire equation by x^3, it restructures the equation so that the left side becomes the derivative of x^3 times y, hence allowing direct integration.

First-Order Differential Equations

First-order differential equations are a class of differential equations where the highest derivative is the first derivative. They often model rates of change in applied problems, appearing in disciplines like physics, engineering, and economics. The general form is:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \(P(x)\) and \(Q(x)\) are functions of \(x\). In order to solve these equations, we employ various methods, including separation of variables, method of integrating factors, and substitution methods.

The integrating factor method is particularly useful when the equation is linear and not separable. By finding an appropriate integrating factor, the first-order linear differential equation becomes ready for direct integration, which is exactly what was done in the exercise provided. Here, the equation \(\frac{dy}{dx} + \frac{3y}{x} = 6x^2\) is made integrable by multiplying by the integrating factor \(x^3\).

Integration

Integration is a fundamental process in calculus that, conversely from differentiation, computes the accumulation of quantities, such as areas under curves, and represents antiderivatives. When we integrate both sides of a differential equation, which is a statement involving derivatives, we're looking for the original function that was differentiated to form the equation.

In the context of differential equations, integration is usually performed after the equation has been simplified or transformed into a form that makes integration straightforward. The resulting expression from performing the integral provides the general solution to the differential equation.

In our exercise, integration was applied after multiplying by the integrating factor:
\[ \int \frac{d}{dx}(x^3y) dx = \int 6x^5 dx \] The integration yields \(x^3y\) on the left side, which is our desired function \(y\) multiplied by the integrating factor, and an antiderivative of \(6x^5\) on the right side, plus the constant of integration, \(C\). Solving for \(y\) gives us the particular solution to the original differential equation.