Partial fractions in differential equations

Use partial fractions to solve a separable differential equation, typically finding a general or particular solution in the form y=f(x).

7 questions · Standard +0.6

1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y)
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CAIE P3 2009 June Q8
10 marks Challenging +1.2
8
  1. Express \(\frac { 100 } { x ^ { 2 } ( 10 - x ) }\) in partial fractions.
  2. Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 100 } x ^ { 2 } ( 10 - x )$$ obtaining an expression for \(t\) in terms of \(x\).
CAIE P3 2017 June Q9
9 marks Standard +0.3
9
  1. Express \(\frac { 1 } { x ( 2 x + 3 ) }\) in partial fractions.
  2. The variables \(x\) and \(y\) satisfy the differential equation $$x ( 2 x + 3 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\). Solve the differential equation and calculate the value of \(y\) when \(x = 9\), giving your answer correct to 3 significant figures.
Edexcel C34 2016 January Q9
10 marks Challenging +1.2
  1. (a) Express \(\frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x - 2 ) }\) in partial fractions.
    (b) Given that \(x > \frac { 2 } { 3 }\), find the general solution of the differential equation
$$x ^ { 2 } ( 3 x - 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y \left( 3 x ^ { 2 } - 4 \right)$$ Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C34 2014 June Q6
10 marks Standard +0.3
6. (a) Express \(\frac { 5 - 4 x } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.
(b) (i) Find a general solution of the differential equation $$( 2 x - 1 ) ( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 - 4 x ) y , \quad x > \frac { 1 } { 2 }$$ Given that \(y = 4\) when \(x = 2\),
(ii) find the particular solution of this differential equation. Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 2007 January Q4
12 marks Standard +0.3
4. (a) Express \(\frac { 2 x - 1 } { ( x - 1 ) ( 2 x - 3 ) }\) in partial fractions.
(b) Given that \(x \geqslant 2\), find the general solution of the differential equation $$( 2 x - 3 ) ( x - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x - 1 ) y$$ (c) Hence find the particular solution of this differential equation that satisfies \(y = 10\) at \(x = 2\), giving your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 2011 January Q3
12 marks Standard +0.3
3. (a) Express \(\frac { 5 } { ( x - 1 ) ( 3 x + 2 ) }\) in partial fractions.
(b) Hence find \(\int \frac { 5 } { ( x - 1 ) ( 3 x + 2 ) } \mathrm { d } x\), where \(x > 1\).
(c) Find the particular solution of the differential equation $$( x - 1 ) ( 3 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 5 y , \quad x > 1$$ for which \(y = 8\) at \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel P4 2022 June Q2
9 marks Standard +0.8
  1. (a) Express \(\frac { 1 } { ( 1 + 3 x ) ( 1 - x ) }\) in partial fractions.
    (b) Hence find the solution of the differential equation
$$( 1 + 3 x ) ( 1 - x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = \tan y \quad - \frac { 1 } { 3 } < x \leqslant \frac { 1 } { 2 }$$ for which \(x = \frac { 1 } { 2 }\) when \(y = \frac { \pi } { 2 }\) Give your answer in the form \(\sin ^ { n } y = \mathrm { f } ( x )\) where \(n\) is an integer to be found.