Combined polynomial and exponential RHS

A question is this type if and only if the right-hand side contains both polynomial and exponential terms requiring separate particular integrals.

3 questions · Standard +0.6

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AQA FP3 2006 June Q1
11 marks Standard +0.3
1 It is given that \(y\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 8 x - 10 - 10 \cos 2 x$$
  1. Show that \(y = 2 x + \sin 2 x\) is a particular integral of the given differential equation.
  2. Find the general solution of the differential equation.
  3. Hence express \(y\) in terms of \(x\), given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
AQA FP3 2011 June Q2
12 marks Standard +0.8
2
  1. Find the values of the constants \(p\) and \(q\) for which \(p + q x \mathrm { e } ^ { - 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 4 - 9 \mathrm { e } ^ { - 2 x }$$
  2. Hence find the general solution of this differential equation.
  3. Hence express \(y\) in terms of \(x\), given that \(y = 4\) when \(x = 0\) and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow 0\) as \(x \rightarrow \infty\).
OCR Further Pure Core 1 2023 June Q5
6 marks Standard +0.8
5
  1. Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + 5 y = 0\).
  2. Hence find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + 5 y = x ( 4 - 5 x )\).