Separable variables

5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y .$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).

6 The variables \(x\) and \(\theta\) satisfy the differential equation $$\sin \frac { 1 } { 2 } \theta \frac { d x } { d \theta } = ( x + 2 ) \cos \frac { 1 } { 2 } \theta$$ for \(0 < \theta < \pi\). It is given that \(x = 1\) when \(\theta = \frac { 1 } { 3 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\cos \theta\).

8 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$ and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 1 } { ( x + 1 ) ( x + 3 ) }$$ It is given that \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

4 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x y } { 1 + x ^ { 2 } }$$ and \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining a simplified expression for \(y\) in terms of \(x\).

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { y - x } ,$$ and \(y = 0\) when \(x = 0\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. Find the value of \(y\) when \(x = 1\), giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.

8 At time \(t\) days after the start of observations, the number of insects in a population is \(N\). The variation in the number of insects is modelled by a differential equation of the form \(\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t\), where \(k\) is a constant and \(N\) is a continuous variable. It is given that when \(t = 0 , N = 100\).

1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
2. Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).
3. Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\cos 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 4 \tan 2 x } { \sin ^ { 2 } 3 y }$$ where \(0 \leqslant x < \frac { 1 } { 4 } \pi\). It is given that \(y = 0\) when \(x = \frac { 1 } { 6 } \pi\).
Solve the differential equation to obtain the value of \(x\) when \(y = \frac { 1 } { 6 } \pi\). Give your answer correct to 3 decimal places.

8

1. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 + 9 y ^ { 2 } } { \mathrm { e } ^ { 2 x + 1 } } .$$ It is given that \(y = 0\) when \(x = 1\).
   Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) as \(x\) tends to infinity. Give your answer in an exact form.

8 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } + 4 } { x ( y + 4 ) }$$ for \(x > 0\). It is given that \(x = 4\) when \(y = 2 \sqrt { 3 }\).
Solve the differential equation to obtain the value of \(x\) when \(y = 2\).

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$ It is given that \(y = 0\) when \(x = 1\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) as \(x\) tends to infinity.

4 The variables \(x\) and \(y\) satisfy the differential equation $$( 1 - \cos x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y \sin x$$ It is given that \(y = 4\) when \(x = \pi\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2 \pi\).

9 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\).
Solve the differential equation and find the exact value of \(y\) when \(x = 3\), giving your answer in a simplified form.

9 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 3 y } \sin ^ { 2 } 2 x$$ It is given that \(y = 0\) when \(x = 0\).
Solve the differential equation and find the value of \(y\) when \(x = \frac { 1 } { 2 }\).

11 The variables \(y\) and \(\theta\) satisfy the differential equation $$( 1 + y ) ( 1 + \cos 2 \theta ) \frac { d y } { d \theta } = e ^ { 3 y }$$ It is given that \(y = 0\) when \(\theta = \frac { 1 } { 4 } \pi\).
Solve the differential equation and find the exact value of \(\tan \theta\) when \(y = 1\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.

8 The coordinates \(( x , y )\) of a general point of a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 1 - 2 x ^ { 2 } \right) y$$ for \(x > 0\). It is given that \(y = 1\) when \(x = 1\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

7 The variables \(x\) and \(t\) satisfy the differential equation $$\mathrm { e } ^ { 3 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } 2 x$$ for \(t \geqslant 0\). It is given that \(x = 0\) when \(t = 0\).

1. Solve the differential equation and obtain an expression for \(x\) in terms of \(t\).
2. State what happens to the value of \(x\) when \(t\) tends to infinity.

7

1. Given that \(y = \ln ( \ln x )\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln x }$$ The variables \(x\) and \(t\) satisfy the differential equation $$x \ln x + t \frac { \mathrm {~d} x } { \mathrm {~d} t } = 0$$ It is given that \(x = \mathrm { e }\) when \(t = 2\).
2. Solve the differential equation obtaining an expression for \(x\) in terms of \(t\), simplifying your answer.
3. Hence state what happens to the value of \(x\) as \(t\) tends to infinity.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 2 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 x y ^ { 2 }$$ and it is given that \(y = 1\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

7 The variables \(x\) and \(\theta\) satisfy the differential equation $$x \sin ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \tan ^ { 2 } \theta - 2 \cot \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\) and \(x > 0\). It is given that \(x = 2\) when \(\theta = \frac { 1 } { 4 } \pi\).

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } \left( \cot ^ { 2 } \theta \right) = - \frac { 2 \cot \theta } { \sin ^ { 2 } \theta }\).
   (You may assume without proof that the derivative of \(\cot \theta\) with respect to \(\theta\) is \(- \operatorname { cosec } ^ { 2 } \theta\).)
2. Solve the differential equation and find the value of \(x\) when \(\theta = \frac { 1 } { 6 } \pi\).

7 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { x } { \tan \theta } \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = x ^ { 2 } + 3$$ It is given that \(x = 1\) when \(\theta = 0\).
Solve the differential equation, obtaining an expression for \(x ^ { 2 }\) in terms of \(\theta\).

11 The variables \(x\) and \(y\) satisfy the differential equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + y = 0$$ It is given that \(x = 1\) when \(y = 1\).

1. Solve the differential equation to obtain an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) when \(x\) tends to infinity. Give your answer in an exact form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 4 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \cos ^ { 2 } 3 y .$$ It is given that \(y = 0\) when \(x = 2\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a car engine, \(t\) minutes after the engine is turned off, is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 15 ) ^ { 2 }$$ where \(k\) is a constant.
Given that the temperature of the car engine

• is \(85 ^ { \circ } \mathrm { C }\) at the instant the engine is turned off
• is \(40 ^ { \circ } \mathrm { C }\) exactly 10 minutes after the engine is turned off
  1. solve the differential equation to show that, according to the model

$$\theta = \frac { a t + b } { c t + d }$$ where \(a , b , c\) and \(d\) are integers to be found.

Hence find, according to the model, the time taken for the temperature of the car engine to reach \(20 ^ { \circ } \mathrm { C }\). Give your answer to the nearest minute.

3．Given that \(\phi = \frac { 1 } { 2 } ( \sqrt { 5 } + 1 )\) ，
（a）show that
（i）\(\phi ^ { 2 } = \phi + 1\)
（ii）\(\frac { 1 } { \phi } = \phi - 1\)
（b）The equations of two curves are $$\begin{array} { r l r l } y & = \frac { 1 } { x } & x > 0 \\ \text { and } & y & = \ln x - x + k & x > 0 \end{array}$$ where \(k\) is a positive constant．
The curves touch at the point \(P\) ．
Find in terms of \(\phi\)
（i）the coordinates of \(P\) ，
（ii）the value of \(k\) ．