@?differential-equations

Flashcards 2022-01-19 What’s the reducible differential equations topic about?? Transforming complicated differential equations into simpler ones using a substitution. If $z = \frac{y}{x}$, or $y = xz$ what is $\frac{\text{d}y}{\text{d}x}$?? $$ \frac{\text{d}y}{\text{d}x} = z + x\frac{\text{d}z}{\text{d}x} $$ What’s the first stage in doing a first-order reducible differential equations question?? Working out how to substitute the derivative by differentiating. If a reducible differential equations question asks you to substitute $z = \frac{y}{x}$ then what is the first step?

See Also Flashcards 2021-09-10 What is the general process for solving a coupled first-order differential equation?? Eliminating one of the variables to form a single second-order differential equation. $$\frac{\text{d}x}{\text{d}t} = x + y \\ \frac{\text{d}y}{\text{d}t} = x - y$$ What are the 3 first steps?? Rewriting it as $y = …$, differentiating and then substituting. $$\frac{\text{d}x}{\text{d}t} = x + y \\ \frac{\text{d}y}{\text{d}t} = x - y$$ You’ve discovered $$x = Ae^{\sqrt{2}t} + Be^{-\sqrt{2}t}$$ What do you now do to find $y$ in terms of $t$?

See Also [[Maths - Integration]]S [[Maths - Differentiation]]S Flashcards What type of answer do you normally get for differential equation questions?? A family of curves. What’s the general process for solving a differential equation question like $$\frac{dy}{dx} = \frac{y + 1}{x}$$?? Seperate the two variables and put them on either side of the equation. Integrate both sides with respect to $x$. Rearrange. $$\int \frac{1}{y + 1} \frac{dy}{dx} dx$$ What does this simplify down to?