Differential Equations

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Q1. Solve the differential equation y' = 5 - 2y.

Solution:

This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).

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Q2. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.

Solution:

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

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Q3. What is the integrating factor for the equation dy/dx + 2y = 3?

Solution:

The integrating factor is e^(∫2dx) = e^(2x).

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Q4. What is the general solution of the equation y'' - 4y' + 4y = 0?

Solution:

The characteristic equation has a repeated root r = 2. The general solution is y = (C1 + C2x)e^(2x).

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Q5. What is the solution to the equation y' + 2y = 0?

Solution:

This is a separable equation. The solution is y = Ce^(-2x).

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Q6. What is the solution of the differential equation y' = 2y + 3?

Solution:

The integrating factor is e^(-2x). Solving gives y = Ce^(2x) + 3/2.

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Q7. What is the solution to the equation dy/dx = y^2? (2022)

Solution:

This is a separable equation. Integrating gives y = 1/(C - x).

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Q8. What is the solution to the equation dy/dx = -5y?

Solution:

This is a separable differential equation. The solution is y = Ce^(-5x), where C is a constant.

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Q9. What is the solution of the equation dy/dx = 4y + 2? (2021)

Solution:

Using an integrating factor, the solution is y = Ce^(4x) - 1/2.

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Q10. Find the solution of the differential equation y' = 3y + 6.

Solution:

This is a linear first-order equation. The integrating factor is e^(3x). The solution is y = Ce^(3x) + 2.