Standard linear first order

1. Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the differential equation

$$\tan x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 3 \cos 2 x \tan x , \quad 0 < x < \frac { \pi } { 2 }$$

4. (i) $$p \frac { \mathrm {~d} x } { \mathrm {~d} t } + q x = r \quad \text { where } p , q \text { and } r \text { are constants }$$ Given that \(x = 0\) when \(t = 0\)

1. find \(x\) in terms of \(t\)
2. find the limiting value of \(x\) as \(t \rightarrow \infty\)
   (ii) $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } + 2 y = \sin \theta$$ Given that \(y = 0\) when \(\theta = 0\), find \(y\) in terms of \(\theta\)
   

7. (a) Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = 2 \cos ^ { 3 } x \sin x + 1 , \quad 0 < x < \frac { \pi } { 2 }$$ Given that \(y = 5 \sqrt { 2 }\) when \(x = \frac { \pi } { 4 }\)
(b) find the value of \(y\) when \(x = \frac { \pi } { 6 }\), giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational numbers to be found.

4. (a) Determine the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = \mathrm { e } ^ { 3 x } \quad x > - 1$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(b) Determine the particular solution of the differential equation for which \(y = 5\) when \(x = 0\)

Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \cot 2 x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(Total 7 marks)

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y = 2 x + 1$$ Find

1. the complementary function,
2. the general solution. In a particular case, it is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
3. Find the solution of the differential equation in this case.
4. Write down the function to which \(y\) approximates when \(x\) is large and positive.

5

1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$ expressing \(y\) in terms of \(x\) in your answer. In a particular case, it is given that \(y = \frac { 2 } { \pi }\) when \(x = \frac { 1 } { 4 } \pi\).
2. Find the solution of the differential equation in this case.
3. Write down a function to which \(y\) approximates when \(x\) is large and positive.

4 Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { x ^ { 2 } y } { 1 + x ^ { 3 } } = x ^ { 2 }$$ for which \(y = 1\) when \(x = 0\), expressing your answer in the form \(y = \mathrm { f } ( x )\).
\(5 \quad\) A line \(l _ { 1 }\) has equation \(\frac { x } { 2 } = \frac { y + 4 } { 3 } = \frac { z + 9 } { 5 }\).

1. Find the cartesian equation of the plane which is parallel to \(l _ { 1 }\) and which contains the points \(( 2,1,5 )\) and \(( 0 , - 1,5 )\).
2. Write down the position vector of a point on \(l _ { 1 }\) with parameter \(t\).
3. Hence, or otherwise, find an equation of the line \(l _ { 2 }\) which intersects \(l _ { 1 }\) at right angles and which passes through the point ( \(- 5,3,4\) ). Give your answer in the form \(\frac { x - a } { p } = \frac { y - b } { q } = \frac { z - c } { r }\).
4. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin x$$
5. Find the solution of the differential equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { 3 }\) when \(x = 0\).

8

1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$ expressing \(y\) in terms of \(x\) in your answer.
2. Find the particular solution for which \(y = 2\) when \(x = \pi\).

1 Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = x ,$$ giving \(y\) in terms of \(x\) in your answer.

3 Use the integrating factor method to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \mathrm { e } ^ { - 3 x }$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \mathrm { f } ( x )\).

1

1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
2. Find the particular solution for which \(y = 1\) when \(x = 0\).

3 Solve the differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = x ^ { 4 } \mathrm { e } ^ { 2 x }\) for \(y\) in terms of \(x\), given that \(y = 0\) when \(x = 1\).

3 Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \cot x = 2 x$$ for which \(y = 2\) when \(x = \frac { 1 } { 6 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).

5 Find the particular solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } + x$$ for which \(y = 1\) when \(x = 1\), giving \(y\) in terms of \(x\).

\begin{enumerate} \item Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = x\) to obtain \(y\) as a function of \(x\). \item (a) Simplify the expression \(\frac { ( x + 3 ) ( x + 9 ) } { x - 1 } - ( 3 x - 5 )\), giving your answer in the form \(\frac { a ( x + b ) ( x + c ) } { x - 1 }\), where \(a , b\) and \(c\) are integers.
(b) Hence, or otherwise, solve the inequality \(\frac { ( x + 3 ) ( x + 9 ) } { x - 1 } > 3 x - 5 \quad\) (4)(Total \(\mathbf { 8 }\) marks) \item (a) Find the general solution of the differential equation \(3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = x ^ { 2 }\)
(b) Find the particular solution for which, at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\).(6)(Total 14 marks) \item The diagram above shows the curve \(C _ { 1 }\) which has polar equation \(\boldsymbol { r } = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \boldsymbol { \operatorname { c o s } } \boldsymbol { \theta } ) , 0 \leq \theta < 2 \pi\) and the circle \(C _ { 2 }\) with equation \(\boldsymbol { r } = \mathbf { 4 } \boldsymbol { a } , 0 \leq \theta < 2 \pi\), where \(a\) is a positive constant.

5 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 x } { x ^ { 2 } + 1 } y = x$$ given that \(y = 1\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).

2

1. Find the values of the constants \(p\) and \(q\) for which \(p \sin x + q \cos x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 5 y = 13 \cos x$$
2. Hence find the general solution of this differential equation.

4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = 2 x ^ { 3 } \mathrm { e } ^ { 2 x }$$ given that \(y = \mathrm { e } ^ { 4 }\) when \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).

4

1. By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = \ln x$$
2. Hence, given that \(y \rightarrow 0\) as \(x \rightarrow 0\), find the value of \(y\) when \(x = 1\).

2

1. Find the values of the constants \(a , b , c\) and \(d\) for which \(a + b x + c \sin x + d \cos x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = 10 \sin x - 3 x$$ (4 marks)
2. Hence find the general solution of this differential equation.

2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sin x$$ given that \(y = 2\) when \(x = 0\).
(9 marks)

4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 3 } { x } y = \left( x ^ { 4 } + 3 \right) ^ { \frac { 3 } { 2 } }$$ given that \(y = \frac { 1 } { 5 }\) when \(x = 1\).
(9 marks)

4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$ given that \(y = \frac { 1 } { 2 }\) when \(x = \frac { \pi } { 6 }\).
(10 marks)

4

1. By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 } { 2 x + 1 } y = 4 ( 2 x + 1 ) ^ { 5 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
2. The gradient of a curve at any point \(( x , y )\) on the curve is given by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 ( 2 x + 1 ) ^ { 5 } - \frac { 4 } { 2 x + 1 } y$$ The point whose \(x\)-coordinate is zero is a stationary point of the curve. Using your answer to part (a), find the equation of the curve.