Standard linear first order - constant coefficients

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

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.

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 \(\theta\)

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.

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.

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

1. Find the values of the constants \(a\), \(b\) and \(c\) for which \(a + b \sin 2 x + c \cos 2 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 4 y = 20 - 20 \cos 2 x$$ [4 marks]
2. Hence find the solution of this differential equation, given that \(y = 4\) when \(x = 0\).
   [0pt] [4 marks]
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Solve the differential equation \(\frac{dy}{dx} - 3y = x\) to obtain \(y\) as a function of \(x\). (Total 5 marks)

Use the integrating factor method to find the solution of the differential equation $$\frac{\text{d}y}{\text{d}x} + 2y = \text{e}^{-3x}$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \text{f}(x)\). [6]

The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} + 4y = 5 \cos 3x.$$

1. Find the complementary function. [2]
2. Hence, or otherwise, find the general solution. [7]
3. Find the approximate range of values of \(y\) when \(x\) is large and positive. [2]