Solve the following differential equations

• dy/dx = x + y

To solve the differential equation dy/dx = x + y, we can use the method of integrating factors or recognize it as a first-order linear ordinary differential equation.

Step 1: Rearranging the Equation

First, we can rewrite the equation in standard form:

dy/dx - y = x

Step 2: Finding the Integrating Factor

The integrating factor, μ(x), is calculated as follows:

μ(x) = e^∫(-1)dx = e^(-x)

Step 3: Multiplying by the Integrating Factor

Now, multiply the entire equation by the integrating factor:

e^(-x) * dy/dx - e^(-x) * y = e^(-x) * x

Step 4: Simplifying the Left Side

The left side can be expressed as the derivative of a product:

d/dx(e^(-x) * y) = e^(-x) * x

Step 5: Integrating Both Sides

Next, integrate both sides:

∫d(e^(-x) * y) = ∫e^(-x) * x dx

Using Integration by Parts

For the right side, we can use integration by parts:

• Let u = x and dv = e^(-x)dx.
• Then, du = dx and v = -e^(-x).

Applying integration by parts gives:

-xe^(-x) - ∫-e^(-x)dx = -xe^(-x) + e^(-x) + C

Step 6: Putting It All Together

Now, we have:

e^(-x) * y = -xe^(-x) + e^(-x) + C

Multiplying through by e^(x) to solve for y:

y = -x + 1 + Ce^(x)

Final Solution

The general solution to the differential equation is:

y = -x + 1 + Ce^(x), where C is a constant determined by initial conditions if provided.