Differential Equations

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Q1. Find the solution of the equation y' + 2y = 0.

Solution:

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

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Q2. 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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Q3. 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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Q4. What is the solution of the differential equation dy/dx = y^2?

Solution:

Separating variables and integrating gives 1/y = x + C, leading to y = 1/(C - x).

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Q5. Find the general solution of the equation y'' - 3y' + 2y = 0.

Solution:

The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r-1)(r-2)=0. Thus, the general solution is y = C1 e^(x) + C2 e^(2x).

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Q6. What is the solution to the differential equation y' = 5y + 3?

Solution:

Using the integrating factor method, we find the solution to be y = (3/5) + Ce^(5x).

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

Solution:

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

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Q8. Solve the differential equation dy/dx = 6x^2y.

Solution:

This is a separable equation. Integrating gives y = Ce^(2x^3).

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Q9. Find the solution of the equation dy/dx = y^2 - 1.

Solution:

This is a separable equation. The solution is y = tan(x + C).

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Q10. Solve the differential equation y'' - 3y' + 2y = 0.

Solution:

The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).