Solve the differential equation dy/dx = 2y + 3. (2023)

₹0.0

Question: Solve the differential equation dy/dx = 2y + 3. (2023)

Options:

Correct Answer: y = Ce^(2x) - 3/2

Exam Year: 2023

Solution:

Using an integrating factor, we find the solution is y = Ce^(2x) - 3/2.

Solve the differential equation dy/dx = 2y + 3. (2023)

Solve the differential equation dy/dx = 2y + 3. (2023)

Step 1: Write the differential equation in standard form: dy/dx - 2y = 3.
Step 2: Identify the integrating factor, which is e^(∫-2dx) = e^(-2x).
Step 3: Multiply the entire equation by the integrating factor: e^(-2x) * dy/dx - 2e^(-2x) * y = 3e^(-2x).
Step 4: The left side of the equation can be rewritten as the derivative of a product: d/dx(e^(-2x) * y) = 3e^(-2x).
Step 5: Integrate both sides with respect to x: ∫d/dx(e^(-2x) * y) dx = ∫3e^(-2x) dx.
Step 6: The left side simplifies to e^(-2x) * y, and the right side integrates to -3/2 * e^(-2x) + C.
Step 7: Solve for y by multiplying both sides by e^(2x): y = Ce^(2x) - 3/2.

First-Order Linear Differential Equations – The question tests the ability to solve a first-order linear differential equation using an integrating factor.
Integrating Factor Method – The solution requires knowledge of how to find and apply an integrating factor to simplify the equation.
Separation of Variables – Understanding that this equation can also be approached through separation of variables, although it's not the primary method indicated.