4.10e Second order non-homogeneous: complementary + particular integral

8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(f ^ { \prime \prime } ( 1 )\).
2. Differentiate ( \(*\) ) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).

10

1. Given that \(y = k x \cos x\) is a particular integral for the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 4 \sin x$$ determine the value of \(k\) and find the general solution of this differential equation.
2. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 5 x - 19$$
   1. Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 1\), find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 1\).
   2. Deduce the Taylor series expansion for \(y\) in ascending powers of \(( x - 1 )\), up to and including the term in \(( x - 1 ) ^ { 3 }\), and use this series to find an approximation correct to 3 decimal places for the value of \(y\) when \(x = 1.1\).

5 Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 24 \mathrm { e } ^ { 2 x }\).

7

1. Find the value of the constant \(k\) for which \(y = k x \sin 2 x\) is a particular integral of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 \cos 2 x\).
2. Solve \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 \cos 2 x\), given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\).

5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 72 \mathrm { e } ^ { 3 x }$$

6

1. Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73 .$$
2. Show that, whatever the initial conditions, \(\frac { y } { x ^ { 2 } } \rightarrow 1\) as \(x \rightarrow \infty\).

Find the general solution of the differential equation $$\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = e^{-\alpha t},$$ where \(\alpha\) is a constant and \(\alpha \neq 2\). [7] Show that if \(\alpha < 2\) then, whatever the initial conditions, \(ye^{\alpha t} \to \frac{1}{(2-\alpha)^2}\) as \(t \to \infty\). [2]

Solve the differential equation $$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 24e^{2x},$$ given that \(y = 1\) and \(\frac{dy}{dx} = 9\) when \(x = 0\). [7]

Find the general solution of the differential equation $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 4x = 7 - 2t^2.$$ [6]

Answer only one of the following two alternatives. **EITHER** It is given that \(w = \cos y\) and $$\tan y \frac{d^2 y}{dx^2} + \left( \frac{dy}{dx} \right)^2 + 2 \tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$

1. Show that $$\frac{d^2 w}{dx^2} + 2 \frac{dw}{dx} + w = -e^{-2x}.$$ [4]
2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0\), \(y = \frac{1}{4}\pi\) and \(\frac{dy}{dx} = \frac{1}{\sqrt{3}}\). [10]

**OR** The curves \(C_1\) and \(C_2\) have polar equations, for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), as follows: \begin{align} C_1 : r &= 2(e^\theta + e^{-\theta}),
C_2 : r &= e^{2\theta} - e^{-2\theta}. \end{align} The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(e^{2\alpha} - 2e^\alpha - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt{2}\). [6]
2. Sketch \(C_1\) and \(C_2\) on the same diagram. [3]
3. Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures. [5]

Find the general solution of the differential equation $$\frac{d^2x}{dt^2} - 8\frac{dx}{dt} - 9x = 9e^{8t}.$$ [6]

It is given that \(y = x^2w\) and $$x^2\frac{d^2w}{dx^2} + 4x(x + 1)\frac{dw}{dx} + (5x^2 + 8x + 2)w = 5x^2 + 4x + 2.$$

1. Show that $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 5x^2 + 4x + 2.$$ [4]
2. Find the general solution for \(w\) in terms of \(x\). [7]

Find the particular solution of the differential equation $$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 3y = 27x^2,$$ given that, when \(x = 0\), \(y = 2\) and \(\frac{dy}{dx} = -8\). [10]

Find the particular solution of the differential equation $$3\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = x^2,$$ given that, when \(x = 0\), \(y = \frac{dy}{dx} = 0\). [10]

$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2e^{-t}.$$ Given that \(x = 0\) and \(\frac{dx}{dt} = 2\) at \(t = 0\),

1. find \(x\) in terms of \(t\). [8]

The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0\), \(P\) is \(x\) metres from the origin \(O\).

1. Show that the maximum distance between \(O\) and \(P\) is \(\frac{2\sqrt{3}}{9}\) m and justify that this distance is a maximum. [7]

1. Find the value of \(z\) for which \(y = zx \sin 5x\) is a particular integral of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [4]
2. Using your answer to part (a), find the general solution of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [3]

Given that at \(x = 0\), \(y = 0\) and \(\frac{dy}{dx} = 5\),

1. find the particular solution of this differential equation, giving your solution in the form \(y = f(x)\). [5]
2. Sketch the curve with equation \(y = f(x)\) for \(0 \leq x \leq \pi\). [2]

\includegraphics{figure_1} The differential equation $$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 9x = \cos 3t, \quad t \geq 0,$$ describes the motion of a particle along the \(x\)-axis.

1. Find the general solution of this differential equation. [8]
2. Find the particular solution of this differential equation for which, at \(t = 0\), \(x = \frac{1}{2}\) and \(\frac{dx}{dt} = 0\). [5]

On the graph of the particular solution defined in part (b), the first turning point for \(T > 30\) is the point \(A\).

1. Find approximate values for the coordinates of \(A\). [2]

Find the general solution of the differential equation $$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2 \cos t - \sin t.$$ [9]

1. Find the value of \(z\) for which \(z^{2e^x}\) is a particular integral of the differential equation $$\frac{d^2 y}{dt^2} - 6 \frac{dy}{dt} + 9y = 6e^{3t}, \quad t \geq 0$$ [5]
2. Hence find the general solution of this differential equation. [3]

Given that when \(t = 0\), \(y = 5\) and \(\frac{dy}{dt} = 4\)

1. find the particular solution of this differential equation, giving your solution in the form \(y = f(t)\). [5]