MATH3503 Online Assignment 11

Make sure to use the link on the Learning Hub, which contains a user name and a user ID unique to you, otherwise your mark would not be recorded. You may use the URL without '?uname=...&uid=...' if you just want to practice. You will get different numbers every time you open the page.
Numerical answers are evaluated as correct if the difference from the correct value is less than `0.001 times` (correct value) unless otherwise specified. That means, you need to keep at least 5 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.

Solve the following differential equations with the given initial or boundary conditions.

\[ \frac{dy}{dx} = 2 + 7x + 4x^2, ~~ y(5) = -2\] `y(x) = `
\[ \frac{dy}{dx} = \frac{3 + 5x + 7x^2}{y^{2}}, ~~ y(0) = 0.4\] `y(x) = `
\[ \frac{dy}{dx} = x^3 y^4, ~~ y(0) = 0.7\] `y(x) = `
\[ \frac{dy}{dx} = x^7 (y+4), ~~ y(0) = 5\] `y(x) = `
* Enter exp(x) for `e^x`
\[ \frac{dy}{dx} + 2 y + 7=0 , ~~ y(0) = 3\] `y(x) = `
\[ \frac{d^2y}{dx^2} = -9 y , ~~ y(0) = 4,~~y'(0) = 5\] `y(x) = `
* Use the ansatz `y = A cos(k x)+B sin(k x)` where `A`, `B` and `k` are constants.
(Newton’s Law of Cooling): Let `T(t)` be the temperature of a hot object cooling down in an environment where the ambient temperature is \(T_\text{env}\). Newton assumed that the rate of cooling is proportional to the temperature difference `T-T_\text{env}`: \[ \frac{dT}{dt}=-k(T-T_\text{env}) \] At `t = 0`, we submerge a hot metal bar with cooling constant `k=2.2` in a large tank of water at temperature \(T_\text{env}=14^\circ\)C. Find the bar’s temperature at time `t` if its initial temperature was \(T(0)=110^\circ\)C. `T(t) = `
The deflection of a beam satisfies \[ \frac{d^2y}{dx^2} = -\frac{1}{49} y , ~~ y(0) = 0,~~y(11) = 0.4\] Find `y(x)`. `y(x) = `