Standard linear first order - variable coefficients

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 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 \(x\).

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 \(y = \mathrm { f } ( x )\).

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 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.

2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \tan ^ { 3 } x \sec x$$ given that \(y = 2\) when \(x = \frac { \pi } { 3 }\).
[0pt] [9 marks]

12 Solve the differential equation \(\left( 4 - x ^ { 2 } \right) \frac { d y } { d x } - x y = 1\), given that \(y = 1\) when \(x = 0\), giving your answer in the form \(y = \mathrm { f } ( x )\).

11 Solve the differential equation \(\cosh x \frac { d y } { d x } - 2 y \sinh x = \cosh x\), given that \(y = 1\) when \(x = 0\).

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

$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = \mathrm { e } ^ { 2 x } \cos ^ { 2 } x$$ giving your answer in the form \(y = \mathrm { f } ( x )\) Given that \(y = 3\) when \(x = 0\) (b) determine the smallest positive value of \(x\) for which \(y = 0\)

3

1. Show that \(y = x ^ { 3 } - x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
2. By differentiating \(\left( x ^ { 2 } - 1 \right) y = c\) implicitly, where \(y\) is a function of \(x\) and \(c\) is a constant, show that \(y = \frac { c } { x ^ { 2 } - 1 }\) is a solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 0$$
3. Hence find the general solution of $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$

3

1. Show that \(x ^ { 2 }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
2. Solve this differential equation, given that \(y = 1\) when \(x = 2\).

3 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$ given that \(y = 3\) when \(x = 0\).

10

1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
2. Find the particular solution for which \(y = 0\) when \(x = 3\) Give your answer in the form \(y = \mathrm { f } ( x )\)

6 Solve the first-order differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 \ln x\) given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).

Find the solution of the differential equation $$\frac{dy}{dx} + \frac{4x^3y}{x^4 + 5} = 6x$$ for which \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f(x)\). [7]

Find the general solution of the differential equation $$\sin x \frac{dy}{dx} - y \cos x = \sin 2x \sin x$$ giving your answer in the form \(y = f(x)\). [8]

Find the general solution of the differential equation $$x \frac{dy}{dx} + 5y = \frac{\ln x}{x}, \quad x > 0,$$ giving your answer in the form \(y = f(x)\). [8]

1. Find, in the form \(y = f(x)\), the general solution of the equation $$\frac{dy}{dx} = 2y \tan x + \sin 2x, \quad 0 < x < \frac{\pi}{2}$$ [6]

Given that \(y = 2\) at \(x = \frac{\pi}{6}\),

1. find the value of \(y\) at \(x = \frac{\pi}{4}\), giving your answer in the form \(a + k \ln b\), where \(a\) and \(b\) are integers and \(k\) is rational. [4]