Further Mathematics · OCR H245 · Topic 4.10

Differential equations

General and particular solutions, modelling, integrating factor method, second order homogeneous differential equations, second order non-homogeneous differential equations, simple harmonic motion, damped oscillations and more.

8 sub-topics
Taught in Year 13
136 practice questions
8 animated lessons

Practice questions (136) Lessons Worksheets coming soon All OCR Further Mathematics topics

What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

4.10aGeneral and particular solutionsY13
4.10bModellingY13
4.10cIntegrating factor methodY13
4.10dSecond order homogeneous differential equationsY13
4.10eSecond order non-homogeneous differential equationsY13
4.10fSimple harmonic motionY13
4.10gDamped oscillationsY13
4.10hLinear systemsY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 136 questions in this topic, free, no account needed.

Differential equations

Not needed for Year 12Easy

Particular solution · Year 13 · 1 mark

The general solution of a differential equation is

y = A\mathrm{e}^{-2x} + 3

.

Find the particular solution with $y = 4$ when $x = 0$ .
(1)

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$A$ ${}= 1$ ${}\Rightarrow y$ ${}= 1\mathrm{e}^{-2x} + 3$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Differential equations

Not needed for Year 12Medium

Integrating factor · Year 13 · 4 marks

Solve $x\frac{\mathrm{d}y}{\mathrm{d}x}$ ${}+ 3y$ ${}= x^2$ , $x > 0$ .
(4)

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Standard form. $\frac{\mathrm{d}y}{\mathrm{d}x}$ ${}+ \frac{3}{x}y$ ${}= x$
$\text{IF}$ ${}= \mathrm{e}^{\int 3/x\,\mathrm{d}x}$ ${}= x^3$
$\frac{\mathrm{d}}{\mathrm{d}x}(x^3y)$ ${}= x^{4}$
$y = \tfrac{1}{5}x^2 + Cx^{-3}$

Total4

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Differential equations

Not needed for Year 12Hard

Second-order non-homogeneous · Year 13 · 4 marks

Find the general solution of $y'' + 5y' + 6y = 4$ .
(4)

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$\text{CF}: y$ ${}= A\mathrm{e}^{-3x}$ ${}+ B\mathrm{e}^{-2x}$
Try a constant. $\text{PI}: y$ ${}= \lambda$ ${}\Rightarrow 6\lambda$ ${}= 4$
$y$ ${}= A\mathrm{e}^{-3x}$ ${}+ B\mathrm{e}^{-2x}$ ${}+ \tfrac{2}{3}$

Total4

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

How did it go?

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