Series solution from differential equation

5. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \left( a x ^ { 2 } + b \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\) and \(b\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\)
3. use your series to estimate the value of \(y\) at \(x = - 0.2\), giving your answer to four decimal places.

4. Given that $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y = 0$$

1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 28 } { y ^ { 2 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 3 } - \frac { 24 } { y } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 8\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients where possible.

1. Given that

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

1. determine \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\) Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
2. determine a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in its simplest form.

1. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \cos x$$

1. Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(x , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). At \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\)
2. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\)
3. Express \(y\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

3. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y - \sin x = 0$$ Given that \(y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 8 }\) at \(x = 0\), find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x ^ { 3 }\).

5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0$$

1. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\). Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0.5\) at \(x = 0\),
2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
   5. \(y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0\)
   1. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\).

5

1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$
2. It is given that \(y = \mathrm { f } ( x )\) is the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$ such that \(\mathrm { f } ( 0 ) = 0\) and \(\mathrm { f } ^ { \prime } ( 0 ) = 0\).
   1. Show that \(f ^ { \prime \prime } ( 0 ) = 0\).
   2. Find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\).
      [0pt] [3 marks]

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 \(x = 1.1\).

A function \(\mathrm{f}(z)\) is defined on all complex numbers \(z\) by \(\mathrm{f}(z) = z^3 - 3z^2 + kz - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm{f}(z) = 0\) are \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha^2 + \beta^2 + \gamma^2 = -5\).

1. Explain why \(\mathrm{f}(z) = 0\) has only one real root. [3]
2. Find the value of \(k\). [3]
3. Find a cubic equation with integer coefficients that has roots \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\) and \(\frac{1}{\gamma}\). [2]