Differential Equations Practice Questions
A-Levels · A-Level Further Mathematics · 149 free MCQs with instant results and detailed explanations.
149
Total
38
Easy
76
Medium
35
Hard
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Sample Questions from Differential Equations
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
If y = e^(2x) is a solution of the differential equation dy/dx - 2y = 0, what type of equation is this?
A.
Homogeneous linear
B.
Non-homogeneous linear
C.
Separable
D.
Exact
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Correct Answer: A
The equation dy/dx - 2y = 0 is a first-order linear differential equation and is homogeneous because it can be expressed in the form of a linear function where all terms are proportional to y.
Q2
Easy
What is the integrating factor for the differential equation dy/dx + 4y = 8?
A.
e^(4x)
B.
e^(-4x)
C.
4e^(4x)
D.
e^(2x)
Show Answer & Explanation
Correct Answer: A
The integrating factor is derived from the coefficient of y in the standard form of the equation. For dy/dx + 4y = 8, the integrating factor is e^(∫4dx) = e^(4x).
Q3
Easy
What is the complementary solution of the differential equation y'' + 5y' + 6y = 0?
A.
y = C1e^(-2x) + C2e^(-3x)
B.
y = C1e^(2x) + C2e^(3x)
C.
y = C1 + C2x
D.
y = C1e^(-x) + C2e^(-6x)
Show Answer & Explanation
Correct Answer: A
To find the complementary solution, solve the characteristic equation associated with the differential equation: r^2 + 5r + 6 = 0, which factors as (r+2)(r+3) = 0, giving roots r = -2 and r = -3. Thus, the complementary solution is y = C1e^(-2x) + C2e^(-3x).
Q4
Medium
Consider the differential equation dy/dx = 3y + 6. What is the general solution?
A.
y = Ce^(3x) - 2
B.
y = 2Ce^(3x)
C.
y = Ce^(-3x) + 2
D.
y = Ce^(6x) - 2
Show Answer & Explanation
Correct Answer: A
The correct solution is derived from separating variables and integrating. The solution form is y = Ce^(3x) - 2 after integrating.
Q5
Medium
What is the particular solution of the differential equation dy/dx + 2y = 4 with the initial condition y(0) = 1?
A.
y = 2 - e^(-2x)
B.
y = 2 + e^(-2x)
C.
y = 4e^(2x)
D.
y = 4 - e^(-2x)
Show Answer & Explanation
Correct Answer: A
The integrating factor method leads to the solution y = 2 - e^(-2x), fitting the initial condition.
Q6
Medium
In the equation dy/dx = y^2 - 1, what is the behavior of the solution as y approaches 1?
A.
It approaches 0
B.
It approaches infinity
C.
It oscillates indefinitely
D.
It remains constant
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Correct Answer: A
As y approaches 1, dy/dx approaches 0. Solutions converge towards the equilibrium point, leading to y approaching 0.
Q7
Medium
Solve the initial value problem: dy/dx = y^2, y(0) = 1.
A.
y = 1/(1-x)
B.
y = 1/(x+1)
C.
y = 1/x
D.
y = 1/(1+x)
Show Answer & Explanation
Correct Answer: A
To solve dy/dx = y^2, we perform separation of variables and integrate. The general solution is y = 1/(C-x). Using the initial condition y(0) = 1, we find C = 1, thus y = 1/(1-x).
Q8
Hard
Consider the differential equation dy/dx + 3y = e^(2x). What is the general solution of this equation?
A.
y = Ce^(-3x) + (1/5)e^(2x)
B.
y = Ce^(3x) - (1/5)e^(2x)
C.
y = Ce^(-3x) - (1/5)e^(2x)
D.
y = Ce^(-3x) + (1/3)e^(2x)
Show Answer & Explanation
Correct Answer: A
To solve the linear differential equation, we use the integrating factor e^(3x). Multiplying through by this gives us eqn in exact form. Integrating leads to the general solution y = Ce^(-3x) + (1/5)e^(2x). Thus, option A is correct.
Q9
Hard
A solution to the differential equation d²y/dx² + 4dy/dx + 4y = 0 is given by y = e^(-2x)(A + Bx). What are the values of A and B if the initial conditions are y(0) = 1 and y'(0) = 0?
A.
A = 1, B = 0
B.
A = 0, B = 1
C.
A = 1, B = 1
D.
A = 0, B = 0
Show Answer & Explanation
Correct Answer: A
Using the initial conditions, we plug in x = 0 into y and y'. From y(0) = 1, we get A = 1. From y'(0) = 0, we can derive that B = 0. Hence, A = 1 and B = 0 is the correct solution.
Q10
Hard
Given the differential equation dy/dx = 3y + 2, what is the general solution?
A.
y = Ce^(3x) - 2/3
B.
y = Ce^(3x) + 2/3
C.
y = 2/3 - Ce^(3x)
D.
y = 2/3 + Ce^(3x)
Show Answer & Explanation
Correct Answer: B
The differential equation is a first-order linear equation. To solve it, we can use the integrating factor method. The integrating factor is e^(3x). After integrating, we obtain the general solution y = Ce^(3x) + 2/3, where C is the constant of integration.
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Differential Equations — A-Levels A-Level Further Mathematics Practice Questions Online
This page contains 149 practice MCQs for the chapter Differential Equations in A-Levels A-Level Further Mathematics. The questions are organized by difficulty — 38 easy, 76 medium, 35 hard — so you can choose the right level for your preparation.
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