Differential Equations

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Q1. What is the solution of the equation y' + 4y = 0?

Solution:

This is a separable equation. The solution is y = Ce^(-4x).

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Q2. What is the solution to the equation y'' - 3y' + 2y = 0?

Solution:

The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2, leading to y = C1 e^(x) + C2 e^(2x).

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Q3. What is the solution to the differential equation dy/dx = -y/x?

Solution:

This is a separable equation. Separating variables and integrating gives y = C/x.

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Q4. What is the solution to the equation y'' + 4y = 0?

Solution:

The characteristic equation is r^2 + 4 = 0, giving complex roots. The general solution is y = C1 cos(2x) + C2 sin(2x).

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Q5. What is the integrating factor for the equation dy/dx + 2y = 6?

Solution:

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

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Q6. 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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Q7. 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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Q8. Find the general solution of dy/dx = 3x^2. (2020)

Solution:

Integrating 3x^2 gives y = x^3 + C.

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Q9. Solve the first-order linear differential equation dy/dx = y/x.

Solution:

This is separable: dy/y = dx/x. Integrating gives ln|y| = ln|x| + C, thus y = Cx.

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Q10. Solve the differential equation dy/dx = 2y + 3. (2023)

Solution:

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