Cambridge Maths Academy

수학 모음 (Maths collection) 전체보기 This little investigation has been motivated by Exercise 7B, Challenge in CP2 §7.2 Second-order homogeneous differential equations. Question 1. What happens with a first-order differential equation with complex constant coefficients? $$ \begin{align} (a_1 + ia_2) \frac{ \textrm{d} y }{ \textrm{d}x } + (b_1 + ib_2) y &= 0 \end{align} $$ Method 1. Using matrix notatio..

Core pure mathematics 2 Table of contents Introduction Examples Exercise 7D 1. Introduction We can use given boundary conditions to find a particular solution to a second-order differential equation. Since there are two arbitrary constants, we will need two boundary conditions to determine the complete particular solution. 2. Examples Example 1. Find $y$ in terms of $x$, given that $$ \begin{ali..

Core pure mathematics 2 Table of contents Complementary function + Particular integral An example Another example Yet another example An investigation: more general cases Exercise 7C 1. Complementary function + Particular integral Second-order differentail equations of the form $$ \begin{align} a \frac{ \textrm{d}^2y }{ \textrm{d}x^2 } + b \frac{ \textrm{d}y }{ \textrm{d}x } + c y = \textrm f(x)..

Core pure mathematics 2 Table of contents First example Arbitrary constants = constants of integration Linear and non-linear differential equations Finding the general solution with the auxiliary equation More examples Exercise 7B 1. First example A second-order differential equation contains second derivatives. Example 1. Find the general solution to the differential equation $$ \begin{align} \..

Core pure mathematics 2 Table of contents First example - the method of separable variables Removing the modulus sign First-order DEs with the method of the integrating factor (I.F.) Second example Two important questions for the integrating factor More examples Exercise 7A First-order differential equations with constant coefficients 1. First example - the method of separable variables If a fir..

수학 모음 (Maths collection) 전체보기 This question comes from a test paper for A-level maths for Michaelmas Term at Eton College. Q16. (a) Solve the equation $$ \begin{align} \log_{10}\left(3^x-5^{2-x}\right)=2+\log_{10}(2)-x\log_{10}(5), \end{align} $$ giving your answer as exact. [3 marks] (b) Solve the equation $$ \begin{align} \left(5\sqrt{5}\right)^{3x}-31=30\left(\frac1{\sqrt[4]{5}}\right)^{9x}, ..

Pure mathematics Year 1 Here's a short introduction to Pure maths 1 Chapter 6 Circles. 6.1 Midpoints and perpendicular bisectors: Find the mid-point of a line segment and the equation of the perpendicular bisector to a line segment. 6.2 Equation of a circle: Learn how to find the equation of a circle. 6.3 Intersections of straight lines and circles: Solve geometric problems involving straight li..