Question

Solve the given differential equations. $$ \frac{d y}{d x}+4 x y=8 x $$

Short answer

The general solution for the given differential equation is: \(y(x) = 2 + \frac{C}{e^{2x^2}}\).

Step-by-step solution

Identify the integrating factor

To find the integrating factor, we look at the coefficient of \(\displaystyle y\) in the given differential equation: \[ \frac{dy}{dx} + 4xy = 8x. \] The coefficient is \(\displaystyle 4x\), which means that the integrating factor is: \[ \mu(x) = e^{\int 4x\, dx}. \]

Calculate the integrating factor

Now we calculate the integrating factor: \[ \mu(x) = e^{\int 4x\, dx} = e^{2x^2}. \]

Multiply both sides by the integrating factor

We multiply both sides of the differential equation by the integrating factor: \[ e^{2x^2}\left(\frac{dy}{dx} + 4xy\right) = 8xe^{2x^2}. \]

Notice that the left-hand side represents the derivative

We can see that the left-hand side is the derivative of the product of the integrating factor and \(\displaystyle y\): \[ \frac{d}{dx}\left(ye^{2x^2}\right) = 8xe^{2x^2}. \]

Integrate both sides and solve for \(\displaystyle y(x)\)

Integrate both sides with respect to \(\displaystyle x\): \[ \int \frac{d}{dx}\left(ye^{2x^2}\right) dx = \int 8xe^{2x^2} dx. \] On the left-hand side, since we are integrating the derivative, the integral simplifies, and we get: \[ ye^{2x^2} = \int 8xe^{2x^2} dx + C, \] where \(\displaystyle C\) is the constant of integration. To solve for \(\displaystyle y( x)\), we simply need to divide both sides of the equation by \(\displaystyle e^{2x^{2}}\): \[ y(x) = \frac{1}{e^{2x^2}}\left(\int 8xe^{2x^2} dx + C\right). \] We can notice that the remaining integral is tricky, so we need to use the substitution method here. Let \(\displaystyle u=2x^{2}\), then \(\displaystyle du=4x\,dx\) and \(\displaystyle x\, dx = \frac{1}{4}du\). The integral now becomes: \[ \int 8xe^{2x^2} dx = \int 8\cdot \frac{1}{4} e^{u}du = 2e^{2x^2} + K, \] for another constant \(\displaystyle K\). Now we can find the final solution: \[ y(x) = \frac{1}{e^{2x^2}}\left(2e^{2x^2} + C\right) = 2 + \frac{C}{e^{2x^2}}. \] So, the general solution for the given differential equation is: \[ y(x) = 2 + \frac{C}{e^{2x^2}}. \]

Key concepts

Integrating Factor

An integrating factor is a function used to simplify the process of solving linear first-order differential equations. It is particularly useful in making a differential equation exact, thereby allowing easier integration.
The integrating factor is generally denoted as \( \mu(x) \). In our exercise, the differential equation is \( \frac{dy}{dx} + 4xy = 8x \), where \( 4x \) is the coefficient of \( y \).
To find the integrating factor, you use the formula \( \mu(x) = e^{\int P(x) \, dx} \), where \( P(x) \) is the coefficient in front of \( y \).
In this case, \( P(x) = 4x \), hence the integrating factor becomes:

• \( \mu(x) = e^{\int 4x \, dx} = e^{2x^2} \).

This function will help in transforming the differential equation into a more manageable form.

Solving Differential Equations

Solving differential equations involves finding a function or set of functions that satisfy the given equation. The solution provides insights into processes modelled by the equation.
In the context of this specific exercise, after calculating the integrating factor \( e^{2x^2} \), we multiply each term of the differential equation by this integrating factor:

• \( e^{2x^2} \left( \frac{dy}{dx} + 4xy \right) = 8xe^{2x^2} \).

The clever choice of an integrating factor helps us turn the left-hand side into a derivative of a product, simplifying the equation's structure and making it solvable by integration.

Derivative

Derivatives play a pivotal role in calculus and differential equations. They describe the rate of change of a function with respect to a variable. Knowing how to manipulate derivatives is essential for solving differential equations.
In our exercise, the equation reformulated with the integrating factor displays the left-hand side as a derivative:

• \( \frac{d}{dx}( ye^{2x^2} ) = 8xe^{2x^2} \).

This equation follows from the rule governing the derivative of a product of functions. The identification and simplification as a derivative allow us to integrate both sides, leading towards the solution. Understanding derivatives and how they behave is crucial to deriving solutions to differential equations.

Substitution Method

The substitution method is a technique used to simplify integration, particularly when dealing with complex expressions. In cases where standard integration appears challenging, substitution can reframe the integral into an easier form.
During the solution of the given problem, substitution was required when integrating the equation:

• \( \int 8xe^{2x^2} \, dx \).

To tackle the complexity, the substitution \( u = 2x^2 \) was employed. This converts the variables and simplifies the integral:

• \( du = 4x \, dx \)
• Transformation: \( 8x \, dx = 2du \)
• New Integral: \( \int 2 e^u \, du \).

Substituting back into the original variable yields the antiderivative needed to solve the differential equation, guiding towards the expression of the general solution.