# 18 Further Pure 3

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

 Maclaurin series Expansions of $e^x, \space \ln(1+x)$,$\cos x$ and $\sin x$, and $(1+x)^n$ for rational values of $n.$ Use of the range of values of $\space x \space$ for which these expansions are valid, as given in the formulae booklet, is expected to determine the range of values for which expansions of related functions are valid; eg. $\ln\Big(\frac{1+x}{1-x}\Big);\space (1-2x)^{\frac{1}{2}}e^x$. Knowledge and use, for $k > 0$, of $\mbox{lim}x^ke^{-x}$ as $x$ tends to infinity and $\mbox{lim} x^k \ln x$ as $\space x \space$ tends to zero. Improper integrals. E.g.$\int_{0}^{e}x\mbox{ln}x \space \mbox{d}x \space, \int^{\infty}_{0}xe^{-x} \space\mbox{d}x$. Candidates will be expected to show the limiting processes used. Use of series expansion to find limits. E.g. $\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} \space ; \space \lim_{x \to 0} \frac{\sin3x}{x} \space ; \space \lim_{x \to 0} \frac{x^2 e^x}{\cos2x-1} \space ; \space \lim_{x \to 0}\frac{\sqrt{2+x}-\sqrt{2}}{x}$
 Relationship between polar and Cartesian coordinates. The convention  $r > 0$  wil be used. The sketching of curves given by equations of the form  $r = \mbox{f}(\theta)$   may be required. Knowledge of the formula $\tan\phi=r{\operatorname{d}\!\theta\over\operatorname{d}\!r}$   is not required. Use of the formula$\mbox{area} = \int_{\alpha}^{\beta}\frac{1}{2}r^2 \space \mbox{d}\theta$.
 The concept of a differential equation and its order. The relationship of order to the number of arbitrary constants in the general solution will be expected. Boundary values and initial conditions, general solutions and particular solutions.
 Analytical solution of first order linear differential equations of the form ${\operatorname{d}\!y\over\operatorname{d}\!x} + \mbox{P}y = \mbox{Q}$ where $P$ and $Q$ are functions of $x$. To include use of an integrating factor and solution by complementary function and particular integral. Numerical methods for the solution of differential eqations of the form ${\operatorname{d}\!y\over\operatorname{d}\!x} = \mbox{f}(x,y)$. Euler's formula and extensions to second order methods for this first order differential equation. Formulae to be used will be stated explicitly in questions, but candidates should be familiar with standard notation such as used in Euler's formula $y_{r+1}=y_{r} + h \mbox{f}(x_r, y_r),$ the formula $y_{r+1} = y_{r-1} + 2h\mbox{f}(x_r, y_r)$, and the formula $y_{r+1}=y_{r}+\frac{1}{2}(k_{1} + k_{2})$ where $k_{1} = h \mbox{f}(x_{r}, y_{r}) \space$ and $\space k_{2} = h \mbox{f}(x_r + h, y_r + k_1)$.
 Solution of differential equations of the form $\displaystyle a{\operatorname{d}^2\!y\over\operatorname{d}\!x^2} + b {\operatorname{d}\!y\over\operatorname{d}\!x}+cy = 0$, where $a$, $b$ and $c$ are integers, by using an auxiliary equation whose roots may be real or complex. Including repeated roots. Solution of equations of the form $\displaystyle a{\operatorname{d}^2\!y\over\operatorname{d}\!x^2} + b {\operatorname{d}\!y\over\operatorname{d}\!x}+cy = \mbox{f}(x)$ where $a$, $b$ and $c$ are integers by finding the complementary function and a particular integral Finding particular integrals will be restricted to cases where $\mbox{f}(x)$ is of the form $e^{kx} \space , \cos kx, \space \sin kx$ or a polynomial of degree at most 4, or a linear combination of any of the above. Solutions of differential equations of the form: $\displaystyle {\operatorname{d}^2\!y\over\operatorname{d}\!x^2} + P{\operatorname{d}\!y\over\operatorname{d}\!x}+Qy = R$ where $P,Q$ and $R$ are functions of $x$. A substitution will always be given which reduces the differential equation to a form which can be directly solved using the other analytical methods in 18.4 and 18.5 of this specification or by separating variables. Level or difficulty as indicated by: (a) Given $\displaystyle x^2{\operatorname{d}^2\!y\over\operatorname{d}\!x^2}-2y = x$ use the substitution $x = e^t$ to show that $\displaystyle {\operatorname{d}^2\!y\over\operatorname{d}\!t^2}-{\operatorname{d}\!y\over\operatorname{d}\!t}-2y = e^t$. Hence find $y$ in terms of $t$ Hence find $y$ in terms of $x$ (b) $\displaystyle (1-x^2){\operatorname{d}^2\!y\over\operatorname{d}\!x^2}-2x{\operatorname{d}\!y\over\operatorname{d}\!x} = 0$  use the subsitution $\displaystyle u={\operatorname{d}\!y\over\operatorname{d}\!x}$ to show that $\displaystyle {\operatorname{d}\!u\over\operatorname{d}\!x} = \frac{2xu}{1-x^2}$ and hence that $u = \frac{A}{1-x^2}$, where $A$ is an arbitrary constant. Hence find $y$ in terms of $x$.