Question

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

Short answer

Use integrating factors to find the solution: \(y = 3x - \frac{\int 3e^{x^2/2} \, dx}{e^{x^2/2}} + Ce^{-x^2/2}\).

Step-by-step solution

Identify the Standard Form

The given differential equation is \(y^{\prime} = 3x + xy\). We start by rewriting it in the standard linear form \(y^{\prime} + P(x)y = Q(x)\), by identifying \(P(x)\) and \(Q(x)\). In this case, \(P(x) = x\) and \(Q(x) = 3x\).

Determine the Integrating Factor

The integrating factor \(\mu(x)\) is determined by the formula \(\mu(x) = e^{\int P(x) \,dx}\). Here, since \(P(x) = x\), we have \[ \mu(x) = e^{\int x \, dx} = e^{x^2/2}. \]

Multiply through by the Integrating Factor

Multiply the entire differential equation by the integrating factor \(e^{x^2/2}\): \[ e^{x^2/2} y^{\prime} + e^{x^2/2} xy = e^{x^2/2} \, 3x. \]

Recognize and Simplify to Derivative of a Product

Notice that the left side of the equation is the derivative of the product \(d/dx(e^{x^2/2} y)\). So we rewrite the equation as: \[ \frac{d}{dx}(e^{x^2/2} y) = 3x e^{x^2/2}. \]

Integrate Both Sides

Integrate both sides of the equation with respect to \(x\): \[ e^{x^2/2} y = \int 3x e^{x^2/2} \, dx. \]This requires integration by parts or substitution.Let \(u = x\) and \(dv = 3e^{x^2/2} \, dx\), \(du = dx\), and integration gives:\[ v = e^{x^2/2}. \]Thus, \[ \int 3x e^{x^2/2} \, dx = 3xe^{x^2/2} - \int 3e^{x^2/2} \, dx. \]

Simplify and Solve for y

The remaining integral \(\int 3e^{x^2/2} \, dx\) does not have a simple antiderivative, so we integrate by parts or numerical integration if needed. Simplifying, we find:\[ e^{x^2/2} y = 3xe^{x^2/2} - \text{(the integral of the second term)} + C. \] Solve for \(y\) by dividing through by \(e^{x^2/2}\):\[ y = 3x - \frac{\int 3e^{x^2/2} \, dx}{e^{x^2/2}} + \frac{C}{e^{x^2/2}}. \]

Write the General Solution

Since the integration of \(\int 3e^{x^2/2} \, dx\) was performed symbolically or numerically, the solution is expressed as:\[ y = 3x - \text{{Solution of the integral}} + Ce^{-x^2/2}. \] This is the general solution for the differential equation.

Key concepts

Integrating Factors

Integrating factors are a powerful tool for solving linear first-order differential equations. They help transform a non-separable differential equation into a simpler form that we can easily integrate. Here's how it works.
First, we need to identify the parts of the differential equation. We put the equation in the standard form:\[\frac{dy}{dx} + P(x)y = Q(x)\]In this scenario, the integrating factor \(\mu(x)\) is computed using the formula:\[\mu(x) = e^{\int P(x) \; dx}\]Multiplying the entire differential equation by this integrating factor modifies the equation into something that can be simplified as the derivative of a product, making integration straightforward.
In our example, we found the integrating factor to be \(e^{x^2/2}\). This integrating factor allows us to simplify the differential equation and solve it through integration.

Linear Differential Equations

A linear differential equation is characterized by its standard format in which a function and its derivatives are linear. This means no powers or products of the function and its derivatives appear in the equation.
Let's recall the standard form of a first-order linear differential equation:\[\frac{dy}{dx} + P(x)y = Q(x)\]In this structure, \(P(x)\) and \(Q(x)\) are functions of \(x\) only. Our exercise posed the equation \(y' = 3x + xy\), which can be rewritten as:\[\frac{dy}{dx} + xy = 3x\]This equation fits the format perfectly where \(P(x) = x\) and \(Q(x) = 3x\). Recognizing and rearranging the equation into this standard form are crucial first steps in solving it using integrating factors.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. This is particularly useful when dealing with expressions that are not easily integrated using standard methods.
The formula for integration by parts is derived from the product rule of differentiation and is given by:\[\int u \; dv = uv - \int v \; du\]In our problem, after simplifying the equation with integrating factors, we needed to compute \(\int 3x e^{x^2/2} \, dx\). By setting \(u = x\) and \(dv = 3e^{x^2/2} \, dx\), the integration becomes manageable by finding the antiderivative of \(v\) and reapplying the integration by parts if necessary.
This method may require repeated application or numerical integration because the resulting integrals might still be complex.

General Solution

The general solution of a differential equation represents a family of solutions, encompassing all possible particular solutions. These solutions satisfy the original equation for any constant \(C\).
Upon solving the linear differential equation using the integrating factors, we are left with an expression that includes a constant \(C\). This constant represents the numerous curves or lines that run parallel to each other and are solutions to the differential equation.
In our example, the general solution is expressed as:\[y = 3x - \frac{\int 3e^{x^2/2} \, dx}{e^{x^2/2}} + Ce^{-x^2/2}\]The constant \(C\) can be adjusted according to initial conditions or boundary values, thereby determining a specific solution tailored to the scenario at hand.