Standard linear first order

1 Find the solution of the differential equation $$\frac { d y } { d x } + 5 y = e ^ { - 7 x }$$ for which \(y = 0\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).

4 Find the solution of the differential equation $$\sin \theta \frac { d y } { d \theta } + y = \tan \frac { 1 } { 2 } \theta$$ where \(0 < \theta < \pi\), given that \(y = 1\) when \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( \theta )\). [You may use without proof the result that \(\int \operatorname { cosec } \theta d \theta = \ln \tan \frac { 1 } { 2 } \theta\).]

5 Find the solution of the differential equation $$x ( x + 7 ) \frac { d y } { d x } + 7 y = x$$ for which \(y = 7\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).

4 Find the solution of the differential equation $$x \frac { d y } { d x } + 2 y = e ^ { x }$$ for which \(y = 3\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).

2 Find the solution of the differential equation $$\frac { d y } { d x } + \frac { 4 x ^ { 3 } y } { x ^ { 4 } + 5 } = 6 x$$ for which \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-04_867_812_278_621} The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) d x < \frac { 4 n ^ { 2 } + 3 n - 1 } { 6 n ^ { 2 } }$$
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) \mathrm { dx }\).

4 Find the solution of the differential equation $$\left( 4 t ^ { 2 } - 1 \right) \frac { d x } { d t } + 4 x = 4 t ^ { 2 } - 1$$ for which \(x = 3\) when \(t = 1\). Give your answer in the form \(\mathrm { x } = \mathrm { f } ( \mathrm { t } )\).

4 Find the solution of the differential equation $$\frac { d y } { d x } + 3 y = \sin x$$ for which \(y = 1\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).

7

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \ln ( \tanh x ) ) = 2 \operatorname { cosech } 2 x\).
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-14_2717_35_106_2015}
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-15_2723_33_99_22}
2. Find the solution of the differential equation $$\sinh 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = \sinh 2 x$$ for which \(y = 5\) when \(x = \ln 2\). Give your answer in an exact form.

3 Fid b sb utiw the id fferen ial eq tin $$x \frac { \mathrm { dy } } { \mathrm { dx } } + 3 y = \frac { \sin x } { x }$$ fo wh ch \(y = O _ { N }\) b \(\mathrm { n } x = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).

6. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 - 2 x ) y = x , \quad x > 0$$ Find the general solution of the differential equation, giving your answer in the form \(y = \mathrm { f } ( x )\).

2. Find, in terms of \(k\), where \(k\) is a positive integer, the general solution of the differential equation

$$( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + k y = x ^ { \frac { 1 } { 2 } } ( 1 + x ) ^ { 2 - k } , \quad x > 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(6)

1. Find the general solution of the differential equation

$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = \left( \cos ^ { 2 } x \right) \ln x , \quad 0 < x < \frac { \pi } { 2 }$$ Give your answer in the form \(y = \mathrm { f } ( x )\).

6. Obtain the general solution of the equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( x \cot x + 2 ) x y = 4 \sin x \quad 0 < x < \pi$$ Give your answer in the form \(y = \mathrm { f } ( x )\)
(8)

1. (a) Determine the general solution of the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y \tan x = \mathrm { e } ^ { 4 x } \sec ^ { 3 } x$$ giving your answer in the form \(y = \mathrm { f } ( x )\)
(b) Determine the particular solution for which \(y = 4\) at \(x = 0\)

6. (a) Find the general solution of the differential equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( \sin x ) y = \cos ^ { 3 } x$$ (b) Show that, for \(0 \leq x \leq 2 \pi\), there are two points on the \(x\)-axis through which all the solution curves for this differential equation pass.
(c) Sketch the graph, for \(0 \leq x \leq 2 \pi\), of the particular solution for which \(y = 0\) at \(x = 0\).

1. (a) Find the general solution of the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = x$$ Given that \(y = 1\) at \(x = 0\),
(b) find the exact values of the coordinates of the minimum point of the particular solution curve,
(c) draw a sketch of this particular solution curve.

5. Find the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \frac { 1 } { x } , \quad x > 0 .$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(7)(Total 7 marks)

1. Obtain the general solution of the differential equation

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = \cos x , \quad x > 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(Total 8 marks)

6. $$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sec ^ { 3 } x$$ Given that \(y = 3\) at \(x = 0\), find \(y\) in terms of \(x\)
(Total 7 marks)

3. Find the general solution of the differential equation $$\sin x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y \cos x = \sin 2 x \sin x$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

3. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = \frac { \ln x } { x } , \quad x > 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

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

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$ Given that \(y = 2\) at \(x = \frac { \pi } { 3 }\)
(b) find the value of \(y\) at \(x = \frac { \pi } { 6 }\), giving your answer in the form \(a + k \ln b\), where \(a\) and \(b\) are integers and \(k\) is rational.

4. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 + x \cot x ) y = \sin x , \quad 0 < x < \pi$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

1. (a) Find the general solution of the differential equation
   (b) Find the particular solution for which \(y = 5\) at \(x = 1\), giving your answer in the form \(y = \mathrm { f } ( x )\).

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x ^ { 2 }$$ (c) (i) Find the exact values of the coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\), making your method clear.
(ii) Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the turning points.

3. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \mathrm { e } ^ { 4 x } \cos ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(b) Find the particular solution for which \(y = 1\) at \(x = 0\)