Differential Equations for Competitive Exam Success

Differential Equations

Q. What is the solution to the differential equation y' = y^2 - 1?

A. y = tan(x + C)
B. y = 1/(C - x)
C. y = C/(1 - Cx)
D. y = C/(x - 1)

Solution

Separating variables and integrating gives y = 1/(C - x).

Correct Answer: B — y = 1/(C - x)

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

A. y = C1 cos(2x) + C2 sin(2x)
B. y = C1 e^(2x) + C2 e^(-2x)
C. y = C1 e^(x) + C2 e^(-x)
D. y = C1 sin(2x) + C2 cos(2x)

Solution

This is a second-order linear homogeneous differential equation. The characteristic equation has roots ±2i, leading to the solution y = C1 cos(2x) + C2 sin(2x).

Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)

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Q. What is the solution to the equation y' = y(1 - y)?

A. y = 1/(C - x)
B. y = C/(C + x)
C. y = C/(1 + Cx)
D. y = C/(1 - Cx)

Solution

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

Correct Answer: B — y = C/(C + x)

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Q. What is the solution to the first-order linear differential equation dy/dx + y = e^x?

A. y = e^x + Ce^(-x)
B. y = e^x - Ce^(-x)
C. y = Ce^x - e^x
D. y = Ce^(-x) + e^x

Solution

Using the integrating factor e^x, we solve to get y = e^x + Ce^(-x).

Correct Answer: A — y = e^x + Ce^(-x)

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Differential Equations

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