Separable Differential Equations and Euler’s Method Practice
Consider the first order differential equation \(\frac{dy}{dx} = y \sin(x)\). (a) For SL students: Solve the differential equation by separation of variables to obtain the general solution. (b) For HL students: Given the initial condition \(y(0)=1\) and using Euler’s method with a step size of \(h=\frac{\pi}{6}\), perform one iteration to approximate \(y\) at \(x=\frac{\pi}{6}\).
Converting Logarithms Between Bases
Using the change-of-base formula, express \(\log_{10}(x)\) in terms of the natural logarithm \(\ln(x)\). Then, given that \(\ln(2) \approx 0.693\) and \(\ln(10) \approx 2.303\), calculate an approximate value for \(\log_{10}(2)\). Show your working clearly.