Standard non-homogeneous with polynomial 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 is a polynomial (e.g., ax² + bx + c).

16 questions · Standard +0.7

4.10e Second order non-homogeneous: complementary + particular integral
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CAIE Further Paper 2 2021 June Q2
7 marks Standard +0.3
2 The variables \(x\) and \(y\) are related by the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 \frac { d y } { d x } + 2 y = 2 x + 1$$
  1. Find the general solution for \(y\) in terms of \(x\).
  2. State an approximate solution for large positive values of \(x\).
CAIE Further Paper 2 2022 June Q3
8 marks Standard +0.8
3 The variables \(t\) and \(x\) are related by the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x } { d t } + x = t ^ { 2 } + 1$$
  1. Find the general solution for \(x\) in terms of \(t\).
  2. Deduce an approximate value of \(\frac { \mathrm { d } ^ { 2 } \mathrm { x } } { \mathrm { dt } ^ { 2 } }\) for large positive values of \(t\).
CAIE Further Paper 2 2023 June Q2
7 marks Standard +0.8
2 The variables \(x\) and \(y\) are related by the differential equation $$6 \frac { d ^ { 2 } x } { d t ^ { 2 } } + 5 \frac { d x } { d t } + x = t ^ { 2 } + 10 t + 13$$
  1. Find the general solution for \(x\) in terms of \(t\).
  2. State an approximate solution for large positive values of \(t\).
CAIE Further Paper 2 2020 November Q2
7 marks Standard +0.8
2 The variables \(x\) and \(y\) are related by the differential equation $$9 \frac { d ^ { 2 } y } { d x ^ { 2 } } + 6 \frac { d y } { d x } + y = 3 x ^ { 2 } + 30 x$$
  1. Find the general solution for \(y\) in terms of \(x\).
  2. State an approximate solution for large positive values of \(x\).
CAIE Further Paper 2 2020 Specimen Q1
6 marks Standard +0.8
1 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = 7 - 2 t ^ { 2 }$$
Edexcel FP2 2006 January Q1
6 marks Moderate -0.3
Find the set of values of \(x\) for which \(\frac { x ^ { 2 } } { x - 2 } > 2 x\).
(Total 6 marks)
CAIE FP1 2012 November Q3
6 marks Standard +0.8
3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26 t ^ { 2 } + 3 t + 13$$
CAIE FP1 2013 November Q3
7 marks Standard +0.8
3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 x ^ { 2 } + 8$$
CAIE FP1 2017 November Q2
6 marks Standard +0.8
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 4 - 5 t ^ { 2 }$$
CAIE FP1 2017 Specimen Q2
6 marks Standard +0.8
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = 7 - 2 t ^ { 2 }$$
OCR MEI FP3 2016 June Q2
24 marks Challenging +1.2
2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).
  1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
  2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
  3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
  4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
  5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).
Pre-U Pre-U 9795/1 Specimen Q6
9 marks Standard +0.8
6
  1. Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73 .$$
  2. Show that, whatever the initial conditions, \(\frac { y } { x ^ { 2 } } \rightarrow 1\) as \(x \rightarrow \infty\).
CAIE FP1 2015 November Q2
6 marks Standard +0.8
Find the general solution of the differential equation $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 4x = 7 - 2t^2.$$ [6]
OCR FP3 Q2
7 marks Standard +0.3
Find the general solution of the differential equation $$\frac{d^2y}{dx^2} - 8\frac{dy}{dx} + 16y = 4x.$$ [7]
Pre-U Pre-U 9795/1 2011 June Q5
7 marks Standard +0.8
Find the general solution of the differential equation \(\frac{d^2 y}{dx^2} + y = 8x^2\). [7]
Pre-U Pre-U 9795/1 2015 June Q9
9 marks Challenging +1.2
The differential equation \((\star)\) is $$\frac{\text{d}^2 u}{\text{d}x^2} + 4u = 8x + 1.$$
  1. Find the general solution of \((\star)\). [5]
  2. The differential equation \((\star \star)\) is $$x \frac{\text{d}^2 v}{\text{d}x^2} + 2 \frac{\text{d}v}{\text{d}x} + 4xv = 8x + 1.$$ By using the substitution \(u = xv\), show that \((\star)\) becomes \((\star \star)\) and deduce the general solution of \((\star \star)\). [4]