Question

Solve the following differential equations by using integrating factors. $$ (x+2) y^{\prime}=3 x+y $$

Short answer

The solution is \( y = 3x(x+2) - 6(x+2)\ln|x+2| + C(x+2) \).

Step-by-step solution

Rearrange the Equation

First, rewrite the original differential equation in the standard linear form, which is \( y' + P(x)y = Q(x) \). The given equation is \((x+2)y' = 3x + y\). Divide every term by \(x+2\) to get \( y' - \frac{1}{x+2}y = \frac{3x}{x+2} \). Now, the equation is in the required linear form with \(P(x) = -\frac{1}{x+2}\) and \(Q(x) = \frac{3x}{x+2}\).

Compute the Integrating Factor

An integrating factor \( \mu(x) \) is given by \( e^{\int P(x) \,dx} \). Here, \( P(x) = -\frac{1}{x+2} \). Therefore, the integrating factor is \( \mu(x) = e^{-\int \frac{1}{x+2} \,dx} \). Integrating gives \( \mu(x) = e^{-\ln|x+2|} = \frac{1}{|x+2|} \).

Multiply through by the Integrating Factor

Multiply every term in the differential equation by the integrating factor \( \mu(x) = \frac{1}{x+2} \) to get \( \frac{y'}{x+2} - \frac{y}{(x+2)^2} = \frac{3x}{(x+2)^2} \). This simplifies to \( \frac{d}{dx}\left(\frac{y}{x+2}\right) = \frac{3x}{(x+2)^2} \), because the left-hand side is the derivative of \( \frac{y}{x+2} \).

Integrate Both Sides

Integrate both sides with respect to \( x \). The left-hand side integrates to \( \frac{y}{x+2} = \int \frac{3x}{(x+2)^2} \,dx \). Use substitution \( u = x+2 \), hence \( du = dx \). Then \( \int \frac{3x}{(x+2)^2} \,dx = \int \frac{3(x+2-2)}{(x+2)^2} \,dx = \int \left(3 - \frac{6}{x+2}\right) \,dx \).

Solve the Integral

The expression \( \int \left(3 - \frac{6}{x+2}\right) \,dx \) integrates to \( 3x - 6\ln|x+2| + C \), where \( C \) is the constant of integration. Thus, \( \frac{y}{x+2} = 3x - 6\ln|x+2| + C \).

Solve for y

Multiply through by \( x+2 \) to solve for \( y \). This gives \( y = (3x - 6\ln|x+2| + C)(x+2) \). Simplify this to obtain the general solution: \( y = 3x(x+2) - 6(x+2)\ln|x+2| + C(x+2) \).

Key concepts

Integrating Factor

An integrating factor is an essential tool in solving linear differential equations. It allows us to transform a non-exact ordinary differential equation into an exact one, which is easier to solve. To find the integrating factor \( \mu(x) \), you first rearrange the differential equation into a standard linear form, \( y' + P(x)y = Q(x) \). Then, the integrating factor is calculated using \( \mu(x) = e^{\int P(x) \,dx} \). Here is why it works:

• It converts the left-hand side of the equation into the derivative of a product.
• By multiplying through by \( \mu(x) \), the equation becomes exact.

In our example, the integrating factor helps to simplify our differential equation, bringing it into a form that allows easier integration. This technique is widely used because it is systematic and applies to a broad class of differential equations.

Linear Differential Equation

When discussing differential equations, a linear differential equation is characterized by terms that are either constants, the dependent variable \( y \), or its derivatives. No terms are multiplied together or raised to a power other than one, and no functions like sine or cosine involve \( y \) or its derivatives. The general form is \( y' + P(x)y = Q(x) \).
It is linear because:

• The dependent variable \( y \) and its derivative \( y' \) appear to the first power.
• They are multiplied by functions of the independent variable \( x \).

In our exercise, the given differential equation was transformed into this linear form after rearranging: \( y' - \frac{1}{x+2} y = \frac{3x}{x+2} \). This allows us to apply the integrating factor method, thus making the solution process manageable.

Integration by Substitution

Integration by substitution is a powerful method used to simplify the process of integration, especially when the integral is complicated. It involves changing variables to transform the integral into an easier form.
To apply substitution, follow these steps:

• Identify a part of the integrand that can be substituted by a single variable \( u \).
• Express \( dx \) in terms of \( du \) by differentiating your substitution.
• Change the limits of integration if definite, or revert back to original variables if indefinite.

In our solution, substitution was used to integrate \( \int \frac{3x}{(x+2)^2} \, dx \). By letting \( u = x+2 \), the integration simplifies, leading to easier integration of terms like \( \int (3 - \frac{6}{u}) \,du \). This method often turns a complex integration task into a basic, more straightforward problem.

Constant of Integration

The constant of integration, often represented as \( C \), is fundamental in indefinite integration. It reflects the fact that there are infinitely many antiderivatives for a given function. Here is why it's crucial:

• When you integrate a function, the derivative loses information about any constants.
• Adding \( C \) restores the general form of the antiderivative.

In practical terms, when solving differential equations, the constant of integration ensures that the solution encompasses all possible particular solutions to the differential equation. In our exercise, after performing the integration, the constant \( C \) appeared as part of the general solution: \( y = (3x - 6\ln|x+2| + C)(x+2) \). It means the solution accounts for all potential initial conditions or specific intervals.