Math Problem Statement

y'=-2y+5

Solution

You have the differential equation:

dydx=2y+5\frac{dy}{dx} = -2y + 5

This is a first-order linear ordinary differential equation. To solve it, we can use the method of separation of variables or integrating factor.

Let's proceed with the integrating factor method. The equation can be rewritten in the form:

dydx+2y=5\frac{dy}{dx} + 2y = 5

This is a linear equation of the form:

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

where P(x)=2P(x) = 2 and Q(x)=5Q(x) = 5. The solution method involves finding the integrating factor μ(x)\mu(x), given by:

μ(x)=eP(x)dx=e2x\mu(x) = e^{\int P(x) \, dx} = e^{2x}

Multiply the entire differential equation by the integrating factor e2xe^{2x}:

e2xdydx+2e2xy=5e2xe^{2x} \frac{dy}{dx} + 2 e^{2x} y = 5 e^{2x}

Now, the left side of the equation is the derivative of e2xye^{2x} y:

ddx(e2xy)=5e2x\frac{d}{dx} \left( e^{2x} y \right) = 5 e^{2x}

Next, integrate both sides with respect to xx:

ddx(e2xy)dx=5e2xdx\int \frac{d}{dx} \left( e^{2x} y \right) \, dx = \int 5 e^{2x} \, dx

The integral of the left-hand side is e2xye^{2x} y, and the integral of the right-hand side is:

5e2xdx=52e2x\int 5 e^{2x} \, dx = \frac{5}{2} e^{2x}

So, we have:

e2xy=52e2x+Ce^{2x} y = \frac{5}{2} e^{2x} + C

Where CC is the constant of integration. Now, divide through by e2xe^{2x}:

y=52+Ce2xy = \frac{5}{2} + Ce^{-2x}

This is the general solution to the differential equation.

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


Here are some related questions to explore:

  1. How does the integrating factor method work for first-order linear differential equations?
  2. What does the constant CC represent in the general solution?
  3. How can we solve this type of equation using separation of variables?
  4. What happens if the right-hand side of the equation is a more complex function (not just a constant)?
  5. Can you solve a similar equation where the coefficient of yy is not constant (e.g., dydx=2xy+5\frac{dy}{dx} = -2xy + 5)?

Tip: When solving differential equations, always check if the equation is linear or separable to choose the best solving method.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
First-Order Linear Differential Equations
Integrating Factor Method

Formulas

dy/dx + P(x)y = Q(x)
Integrating Factor: μ(x) = e^{∫P(x)dx}
General Solution: y = (1/μ(x)) ∫(μ(x) Q(x) dx) + C/μ(x)

Theorems

Integrating Factor Theorem

Suitable Grade Level

Grades 11-12, College Calculus