Standard non-homogeneous with trigonometric RHS

A question is this type if and only if it asks to solve a second-order linear differential equation with constant coefficients where the right-hand side contains sin or cos terms (not involving resonance).

10 questions · Standard +0.8

4.10e Second order non-homogeneous: complementary + particular integral
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Edexcel FP2 2012 June Q4
9 marks Standard +0.8
4. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \cos t - \sin t$$
OCR FP3 2009 January Q4
9 marks Standard +0.8
4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
OCR FP3 2015 June Q1
8 marks Standard +0.8
1 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = \sin x$$
OCR FP3 2016 June Q5
8 marks Standard +0.8
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 85 \cos x .$$
CAIE FP1 2014 June Q4
6 marks Standard +0.8
4 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 195 \sin 2 t$$
AQA FP3 2010 June Q2
7 marks Standard +0.8
2
  1. Find the value of the constant \(k\) for which \(k \sin 2 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = \sin 2 x$$
  2. Hence find the general solution of this differential equation.
OCR Further Pure Core 1 2018 March Q6
6 marks Moderate -0.5
6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point \(O\). Initially the mass hangs at rest vertically below \(O\). The mass is then pulled to one side with the string taut and released from rest. \(\theta\) is the angle, in radians, that the string makes with the vertical through \(O\) at time \(t\) seconds and \(\theta\) may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$
  1. Write down the general solution to this differential equation.
  2. Initially the pendulum is released from rest at an angle of \(\theta _ { 0 }\). Find the particular solution to the equation in this case.
  3. State any limitations on the model.
Edexcel FP2 Q4
9 marks Standard +0.8
Find the general solution of the differential equation $$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2 \cos t - \sin t.$$ [9]
OCR FP3 Q8
13 marks Challenging +1.8
  1. Find the complementary function of the differential equation $$\frac{d^2y}{dx^2} + y = \cosec x.$$ [2]
  2. It is given that \(y = p(\ln \sin x) \sin x + qx \cos x\), where \(p\) and \(q\) are constants, is a particular integral of this differential equation.
    1. Show that \(p - 2(p + q) \sin^2 x \equiv 1\). [6]
    2. Deduce the values of \(p\) and \(q\). [2]
  3. Write down the general solution of the differential equation. State the set of values of \(x\), in the interval \(0 \leqslant x \leqslant 2\pi\), for which the solution is valid, justifying your answer. [3]
OCR FP3 Q4
9 marks Standard +0.8
Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 65 \sin 2x.$$ [9]