Math Problem Statement

dy/dx=-1-3y

Solution

The given differential equation is:

dydx=13y\frac{dy}{dx} = -1 - 3y

This is a first-order linear differential equation. To solve it, we will use the method of separation of variables or an integrating factor. Let's proceed with the integrating factor method.

Step 1: Rewrite the equation

dydx+3y=1\frac{dy}{dx} + 3y = -1

This is now in the standard linear form:

dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)

where P(x)=3P(x) = 3 and Q(x)=1Q(x) = -1.

Step 2: Find the integrating factor

The integrating factor μ(x)\mu(x) is given by:

μ(x)=eP(x)dx=e3dx=e3x\mu(x) = e^{\int P(x) dx} = e^{\int 3 dx} = e^{3x}

Step 3: Multiply the equation by the integrating factor

Multiplying both sides of the equation by e3xe^{3x}, we get:

e3xdydx+3e3xy=e3xe^{3x} \frac{dy}{dx} + 3e^{3x} y = -e^{3x}

The left-hand side is now the derivative of e3xye^{3x} y:

ddx(e3xy)=e3x\frac{d}{dx} \left( e^{3x} y \right) = -e^{3x}

Step 4: Integrate both sides

Integrating both sides with respect to xx:

e3xy=e3xdxe^{3x} y = \int -e^{3x} dx

The integral on the right side is:

e3xdx=e3x3\int -e^{3x} dx = \frac{-e^{3x}}{3}

Thus, we have:

e3xy=e3x3+Ce^{3x} y = \frac{-e^{3x}}{3} + C

Step 5: Solve for yy

To solve for yy, divide both sides by e3xe^{3x}:

y=e3x3e3x+Ce3x=13+Ce3xy = \frac{-e^{3x}}{3e^{3x}} + \frac{C}{e^{3x}} = -\frac{1}{3} + Ce^{-3x}

Thus, the general solution to the differential equation is:

y=13+Ce3xy = -\frac{1}{3} + Ce^{-3x}

where CC is the constant of integration.


Would you like more details on any step or have any questions?

Here are some related questions for further exploration:

  1. What is the general form of a linear differential equation?
  2. How does the integrating factor method work for solving differential equations?
  3. Can this differential equation be solved using separation of variables?
  4. What are the conditions for a differential equation to be classified as first-order linear?
  5. How can initial conditions be applied to determine the constant CC?

Tip: When dealing with first-order linear differential equations, the integrating factor method is a reliable approach to finding the solution!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Differential Equations
First-order Linear Equations
Integrating Factor

Formulas

First-order linear form: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^∫P(x)dx

Theorems

Integrating Factor Method

Suitable Grade Level

Undergraduate level (Calculus I or II)