Differential equations questions

Questions to attempt

6 questions on Differential equations, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-9.1 · Find and use an integrating factor to solve differential equations of form dy/dx + P(x)y = Q(x) and recognise when it is appropriate to do so.

   Q1

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   1.

   For $x>0$, solve $\dfrac{dy}{dx}+\dfrac2x y=x^2$.

   (5)

   (Total for Question 1 is 5 marks)

   Reveal answer and method

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2. CP-9.2 · Find both general and particular solutions to differential equations.

   Q2

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   2.

   Solve $y''-y'=0$ subject to $y(0)=2$ and $y'(0)=-3$.

   (5)

   (Total for Question 2 is 5 marks)

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3. CP-9.3 · Use differential equations in modelling in kinematics and in other contexts.

   Q3

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   3.

   A particle falls from rest. Taking downward as positive, its speed satisfies $\dfrac{dv}{dt}=10-0.5v$. Find $v(t)$ and the terminal speed predicted by the model.

   (5)

   (Total for Question 3 is 5 marks)

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4. CP-9.4 · Solve differential equations of form y'' + ay' + by = 0 where a and b are constants by using the auxiliary equation.

   Q4

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   4.

   Find the general solution of $y''-6y'+9y=0$.

   (4)

   (Total for Question 4 is 4 marks)

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5. CP-9.5 · Solve differential equations of form y'' + ay' + by = f(x) by solving the homogeneous case and adding a particular integral to the complementary function (f(x) polynomial, exponential or trigonometric).

   Q5

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   5.

   Find the general solution of $y''-4y=8e^{2x}$.

   (5)

   (Total for Question 5 is 5 marks)

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6. CP-9.6 · Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation.

   Q6

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   6.

   Given $k>0$, find the value of $k$ for which $y''+ky'+16y=0$ has a repeated auxiliary root. Write the corresponding general solution.

   (4)

   (Total for Question 6 is 4 marks)

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CP-9 Differential equations revision notes