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

7 The variables \(x\) and \(y\) are related by the differential equation $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 3 } = 8 \mathrm { e } ^ { - x }$$ Given that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 15 v = 24 \mathrm { e } ^ { - x }$$ Hence find the general solution for \(y\) in terms of \(x\).

8 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t$$ Find the particular solution, given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\) when \(t = 0\). State an approximate solution for large positive values of \(t\).

8 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 10 \mathrm { e } ^ { - 2 x }$$ given that \(y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\).

Show that the substitution \(y = x z\) reduces the differential equation $$\frac { 1 } { x } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { 6 } { x } - \frac { 2 } { x ^ { 2 } } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + \left( \frac { 9 } { x } - \frac { 6 } { x ^ { 2 } } + \frac { 2 } { x ^ { 3 } } \right) y = 169 \sin 2 x$$ to the differential equation $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} z } { \mathrm {~d} x } + 9 z = 169 \sin 2 x$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , z = - 10\) and \(\frac { \mathrm { d } z } { \mathrm {~d} x } = 5\).

9 Find \(x\) in terms of \(t\) given that $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 6 \mathrm { e } ^ { - 2 t }$$ and that, when \(t = 0 , x = \frac { 5 } { 3 }\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 7 } { 6 }\). State \(\lim _ { t \rightarrow \infty } x\).

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 \(x = 0\).

4 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 195 \sin 2 t$$

10 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 0.16 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 0.0064 x = 8.64 + 0.32 t$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\). Show that, for large positive \(t , \frac { \mathrm {~d} x } { \mathrm {~d} t } \approx 50\).

Show that the substitution \(v = \frac { 1 } { y }\) reduces the differential equation $$\frac { 2 } { y ^ { 3 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - \frac { 1 } { y ^ { 2 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { 2 } { y ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \frac { 5 } { y } = 17 + 6 x - 5 x ^ { 2 }$$ to the differential equation $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 5 v = 17 + 6 x - 5 x ^ { 2 }$$ Hence find \(y\) in terms of \(x\), given that when \(x = 0 , y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 1\).

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 \(x = 0\).

10 It is given that \(x = t ^ { \frac { 1 } { 2 } }\), where \(x > 0\) and \(t > 0\), and \(y\) is a function of \(x\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 t ^ { \frac { 1 } { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 t \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Hence show that the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 8 x + \frac { 1 } { x } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 12 x ^ { 2 } y = 4 x ^ { 2 } \mathrm { e } ^ { - x ^ { 2 } }$$ reduces to the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = \mathrm { e } ^ { - t }$$
3. Find the general solution of ( \(*\) ), giving \(y\) in terms of \(x\).

8 Find the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 18 t ^ { 2 } + 6 t + 1$$ given that, when \(t = 0 , x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).

7 Find the particular solution of the differential equation $$49 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 14 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 49 x + 735$$ given that when \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

10 It is given that \(t \neq 0\) and $$t \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 t x = 3 t ^ { 2 } + 1$$

1. Show that if \(y = t x\) then $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 9 y = 3 t ^ { 2 } + 1$$
2. Find \(x\) in terms of \(t\), given that \(x = \frac { 1 } { 9 } \pi\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 }\) when \(t = \frac { 1 } { 3 } \pi\).

7 Find the particular solution of the differential equation $$10 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 3 \frac { \mathrm {~d} x } { \mathrm {~d} t } - x = t + 2$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).

8 Find the particular solution of the differential equation $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 50 \sin t$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).

8 The value of the assets of a large commercial organisation at time \(t\), measured in years, is \(\\) \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)\(. The variables \)y\( and \)t$ are related by the differential equation $$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$ Find \(y\) in terms of \(t\), given that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2\) when \(t = 0\). Show that, for large values of \(t\), the value of the assets is less than \(\\) 9.5 \times 10 ^ { 8 }$ for about a third of the time.

The variable \(y\) depends on \(x\), and the variables \(x\) and \(t\) are related by \(x = \mathrm { e } ^ { t }\). Show that $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t } \quad \text { and } \quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } .$$

1. Given that \(y\) satisfies the differential equation $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 50 ( \ln x ) - 1$$ find a differential equation involving only \(t\) and \(y\).
2. Show that the complementary function of the differential equation in \(t\) and \(y\) may be written in the form $$R \mathrm { e } ^ { - \frac { 3 } { 2 } t } \sin ( 2 t + \phi )$$ where \(R\) and \(\phi\) are arbitrary constants.
3. Find a particular integral of the differential equation in \(t\) and \(y\).
4. Hence find the general solution of the differential equation in \(x\) and \(y\).

8 Given that $$2 y ^ { 3 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 12 y ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y ^ { 2 } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 17 y ^ { 4 } = 13 \mathrm { e } ^ { - 4 x }$$ and that \(v = y ^ { 4 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 34 v = 26 \mathrm { e } ^ { - 4 x }$$ Hence find the general solution for \(y\) in terms of \(x\).

8 Find \(y\) in terms of \(t\), given that $$5 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 5 y = 15 + 12 t + 5 t ^ { 2 }$$ and that \(y = \frac { \mathrm { d } y } { \mathrm {~d} t } = 0\) when \(t = 0\).

9 Show that if \(y\) depends on \(x\) and \(x = \mathrm { e } ^ { u }\) then $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} u } .$$ Given that \(y\) satisfies the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 30 x ^ { 2 }$$ use the substitution \(x = \mathrm { e } ^ { u }\) to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 3 y = 30 \mathrm { e } ^ { 2 u }$$ Hence find the general solution for \(y\) in terms of \(x\).

11 It is given that \(x \neq 0\) and $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 x y = 8 x ^ { 2 } + 16$$ Show that if \(z = x y\) then $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 4 z = 8 x ^ { 2 } + 16$$ Find \(y\) in terms of \(x\), given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2\) when \(x = \frac { 1 } { 2 } \pi\).

6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = \sin 2 t$$ Describe the behaviour of \(x\) as \(t \rightarrow \infty\), justifying your answer.

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26 t ^ { 2 } + 3 t + 13$$

Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\).