4.10c Integrating factor: first order equations

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 \(x = 0\)

7. (a) Show that the substitution \(v = y ^ { - 3 }\) transforms the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 2 x ^ { 4 } y ^ { 4 }$$ into the differential equation $$\begin{aligned} & \frac { \mathrm { d } v } { \mathrm {~d} x } - \frac { 3 v } { x } = - 6 x ^ { 3 } \\ & \text { ration (II), find a general solution of differential equation (I) } \end{aligned}$$ in the form \(y ^ { 3 } = \mathrm { f } ( x )\).

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.

8. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find $$\int 2 x ^ { 5 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ (b) Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 x ^ { 2 } \mathrm { e } ^ { - x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). Given that \(y = 0\) when \(x = 1\) (c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).

7. A population \(P\) is growing at a rate which is modelled by the differential equation $$\frac { d P } { d t } - 0.1 P = 0.05 t$$ where \(t\) years is the time that has elapsed from the start of observations.
It is given that the population is 10000 at the start of the observations.

1. Solve the differential equation to obtain an expression for \(P\) in terms of \(t\).
2. Show that the population doubles between the sixth and seventh year after the observations began.
   (2)

5. (a) Obtain the general solution of the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } - 0.1 S = t$$ (b) The differential equation in part (a) is used to model the assets, \(\pounds S\) million, of a bank \(t\) years after it was set up. Given that the initial assets of the bank were \(\pounds 200\) million, use your answer to part (a) to estimate, to the nearest \(\pounds\) million, the assets of the bank 10 years after it was set up.

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 \(x = 0\)

1. (a) Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 x y = x \mathrm { e } ^ { - x ^ { 2 } } y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 x z = - 2 x \mathrm { e } ^ { - x ^ { 2 } }$$ (b) Solve differential equation (II) to find \(z\) as a function of \(x\).
(c) Hence find the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

1. (a) Show that the transformation \(z = y ^ { \frac { 1 } { 2 } }\) transforms the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } - 4 y \tan x = 2 y ^ { \frac { 1 } { 2 } }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$ (b) Solve the differential equation (II) to find \(z\) as a function of \(x\).
(c) Hence obtain the general solution of the differential equation (I). $$\left[ \begin{array} { l } \text { Leave } \\ \text { blank } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \end{array} \right.$$

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \left( 1 + \frac { 3 } { x } \right) = \frac { 1 } { x ^ { 2 } } , \quad x > 0$$

1. Verify that \(x ^ { 3 } \mathrm { e } ^ { x }\) is an integrating factor for the differential equation.
2. Find the general solution of the differential equation.
3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\).
   (3)(Total 10 marks)

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)

7

1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).

7

1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).

4 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$

1. Use the substitution \(y = x z\), where \(z\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
2. Hence show by integration that the general solution of the differential equation (A) may be expressed in the form \(x ^ { 2 } \left( x ^ { 2 } - 2 y ^ { 2 } \right) = k\), where \(k\) is a constant.

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.

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.

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

$$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + x y - x = 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(b) Find the particular solution for which \(y = 2\) when \(x = 3\)

1 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y } .$$

1. Use the substitution \(y = u x\), where \(u\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } = \frac { 2 } { u } .$$
2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

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 \(x\).