Resonance cases requiring modified PI

8. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 4 \mathrm { e } ^ { 3 t } , \quad t \geq 0 .$$

1. Show that \(K t ^ { 2 } e ^ { 3 t }\) is a particular integral of the differential equation, where \(K\) is a constant to be found.
2. Find the general solution of the differential equation. (3) Given that a particular solution satisfies \(\boldsymbol { y } = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) when \(\boldsymbol { t } = \mathbf { 0 }\),
3. find this solution.(4) Another particular solution which satisfies \(\boldsymbol { y } = \mathbf { 1 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = \mathbf { 0 }\) when \(\boldsymbol { t } = \mathbf { 0 }\), has equation $$y = \left( 1 - 3 t + 2 t ^ { 2 } \right) \mathrm { e } ^ { 3 t }$$
4. For this particular solution draw a sketch graph of \(y\) against \(t\), showing where the graph crosses the \(t\)-axis. Determine also the coordinates of the minimum of the point on the sketch graph.

8. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ (b) Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
(d) Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\).

1. (a) Find the value of \(\lambda\) for which \(\lambda t ^ { 2 } \mathrm { e } ^ { 3 t }\) is a particular integral of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 6 \mathrm { e } ^ { 3 t } , \quad t \geqslant 0$$ (b) Hence find the general solution of this differential equation. Given that when \(t = 0 , y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 4\) (c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( t )\).

7. (a) Find the value of the constant \(\lambda\) for which \(y = \lambda x \mathrm { e } ^ { 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 y = 6 \mathrm { e } ^ { 2 x }$$ (b) Hence, or otherwise, find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 y = 6 \mathrm { e } ^ { 2 x }$$

9. Resonance in an electrical circuit is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } } + 64 V = \cos 8 t$$ where \(V\) represents the voltage in the circuit and \(t\) represents time.

1. Find the value of \(\lambda\) for which \(\lambda\) tsin8t is a particular integral of the differential equation.
2. Find the general solution of the differential equation. Given that \(V = 0\) and \(\frac { \mathrm { d } V } { \mathrm {~d} t } = 0\) when \(t = 0\),
3. find the particular solution of the equation.
4. Describe the behaviour of \(V\) as \(t\) becomes large, according to this model.

1. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ (b) Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
(d) Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\).

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 y = 8 \cos 4 x$$

1. Find the complementary function of the differential equation.
2. Given that there is a particular integral of the form \(y = p x \sin 4 x\), where \(p\) is a constant, find the general solution of the equation.
3. Find the solution of the equation for which \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).

5 The variables \(x\) and \(y\) satisfy the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 5 \mathrm { e } ^ { - 2 x }$$

1. Find the complementary function of the differential equation.
2. Given that there is a particular integral of the form \(y = p x \mathrm { e } ^ { - 2 x }\), find the constant \(p\).
3. Find the solution of the equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\).

6 The differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin k x\) is to be solved, where \(k\) is a constant.

1. In the case \(k = 2\), by using a particular integral of the form \(a x \cos 2 x + b x \sin 2 x\), find the general solution.
2. Describe briefly the behaviour of \(y\) when \(x \rightarrow \infty\).
3. In the case \(k \neq 2\), explain whether \(y\) would exhibit the same behaviour as in part (ii) when \(x \rightarrow \infty\).

5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = \mathrm { e } ^ { 3 x }$$

1. Find the complementary function.
2. Explain briefly why there is no particular integral of either of the forms \(y = k \mathrm { e } ^ { 3 x }\) or \(y = k x \mathrm { e } ^ { 3 x }\).
3. Given that there is a particular integral of the form \(y = k x ^ { 2 } \mathrm { e } ^ { 3 x }\), find the value of \(k\).

7 Find the value of the constant \(\lambda\) such that \(\lambda x \mathrm { e } ^ { - x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 6 \mathrm { e } ^ { - x }$$ Find the solution of the differential equation for which \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) when \(x = 0\).

9 Find the value of the constant \(k\) such that \(y = k x ^ { 2 } \mathrm { e } ^ { 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 \mathrm { e } ^ { 2 x }$$ Hence find the general solution of ( \(*\) ). Find the particular solution of ( \(*\) ) such that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2\) when \(x = 0\).

5 It is given that \(y\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \mathrm { e } ^ { - 2 x }$$

1. Find the value of the constant \(p\) for which \(y = p x \mathrm { e } ^ { - 2 x }\) is a particular integral of the given differential equation.
2. Solve the differential equation, expressing \(y\) in terms of \(x\), given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).

3 It is given that the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$ is \(y = \mathrm { e } ^ { x } ( A x + B )\). Hence find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 6 \mathrm { e } ^ { x }$$

10. \begin{figure}[h] \end{figure} The motion of a pendulum, shown in Figure 3, is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ where \(\theta\) is the angle, in radians, that the pendulum makes with the downward vertical, \(t\) seconds after it begins to move.

1. 1. Show that a particular solution of the differential equation is $$\theta = \frac { 1 } { 12 } t \sin 3 t$$
   2. Hence, find the general solution of the differential equation. Initially, the pendulum
      • makes an angle of \(\frac { \pi } { 3 }\) radians with the downward vertical
2. is at rest
Given that, 10 seconds after it begins to move, the pendulum makes an angle of \(\alpha\) radians with the downward vertical,
3. determine, according to the model, the value of \(\alpha\) to 3 significant figures. Given that the true value of \(\alpha\) is 0.62
4. evaluate the model. The differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ models the motion of the pendulum as moving with forced harmonic motion.
5. Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

15. (a) Find the value of \(\lambda\) for which \(\lambda x \cos 3 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 y = - 12 \sin 3 x$$ (b) Hence find the general solution of this differential equation. The particular solution of the differential equation for which \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(x = 0\), is \(y = \mathrm { g } ( x )\).
(c) Find \(\mathrm { g } ( x )\).
(d) Sketch the graph of \(y = \mathrm { g } ( x ) , 0 \leq x \leq \pi\).
[0pt] [P4 January 2003 Qn 7]

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1. Show that Charlotte's method will fail to find a particular integral for the differential equation.
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2. Explain how Charlotte should have started her solution differently and find the general solution of the differential equation.