# linear differential equations? (1 Viewer)

#### indeed

##### Active Member
(from the James Ruse 2023 3 u paper)

Do you guys think this is necessary to know, the working involves using the below equation which i never learnt

#### Luukas.2

##### Well-Known Member
(from the James Ruse 2023 3 u paper)
\bg_white \begin{align*} \text{A first order DE is linear in y if it can be written as} \qquad f(x) \times \cfrac{dy}{dx} + g(x) \times y &= h(x) \qquad \text{where f(x), g(x), and h(x) are not restricted as to degree} \\ \\ \text{Option (A) can be written as} \qquad 1 \times \cfrac{dy}{dx} + 0 \times y &= 2x^2 \qquad \text{and so fits the form to be linear in y} \\ \text{Option (B) can be written as} \qquad 1 \times \cfrac{dy}{dx} + -1 \times y &= x \qquad \text{and so fits the form to be linear in y} \\ \text{Option (D) can be written as} \qquad x \times \cfrac{dy}{dx} + -3 \times y &= 0 \qquad \text{and so fits the form to be linear in y} \\ \\ \text{Option (C) can be written as} \qquad 1 \times \cfrac{dy}{dx} + -3 \times y^2 &= 0 \qquad \text{but this is \emph{not} linear in y} \\ \text{as writing it as} \qquad 1 \times \cfrac{dy}{dx} + -3y \times y &= 0 \qquad \text{requires g(x,\ \!y) = -3y, a function of x and y.} \end{align*}