Solution for ∫y(x)dx = ky, = k/y, = kx, = k/x

[Jhenrique]

Homework Statement

Get the solution (for y(x)) for the follows integrals: $$\int y(x) dx = ky\;\;\;\;\;(1)$$ $$\int y(x) dx = \frac{k}{y}\;\;\;\;\;(2)$$ $$\int y(x) dx = kx\;\;\;\;\;(3)$$ $$\int y(x) dx = \frac{k}{x}\;\;\;\;\;(4)$$

Homework Equations

The Attempt at a Solution

$$\\ \int y(x)dx = kx \\ \\ \int y(x)\frac{dx}{dx} = \frac{kx}{dx} \\ \\ d\int y(x) = d\frac{kx}{dx} \\ \\ y(x) = k\frac{dx}{dx} \\ \\ y(x) = k$$

$$\\ \int y(x)dx = \frac{k}{x} \\ \\ \int y(x)\frac{dx}{dx} = \frac{1}{dx} \frac{k}{x} \\ \\ d\int y(x) = \frac{d}{dx} \frac{k}{x} \\ \\ y(x) = k \frac{d}{dx}\left ( \frac{1}{x} \right ) \\ \\ y(x) = -\frac{k}{x^2}$$

 

[Mark44]

Homework Statement

Get the solution (for y(x)) for the follows integrals: $$\int y(x) dx = ky\;\;\;\;\;(1)$$ $$\int y(x) dx = \frac{k}{y}\;\;\;\;\;(2)$$ $$\int y(x) dx = kx\;\;\;\;\;(3)$$ $$\int y(x) dx = \frac{k}{x}\;\;\;\;\;(4)$$

Homework Equations

The Attempt at a Solution

$$\\ \int y(x)dx = kx \\ \\ \int y(x)\frac{dx}{dx} = \frac{kx}{dx} \\ \\ d\int y(x) = d\frac{kx}{dx} \\ \\ y(x) = k\frac{dx}{dx} \\ \\ y(x) = k$$

$$\\ \int y(x)dx = \frac{k}{x} \\ \\ \int y(x)\frac{dx}{dx} = \frac{1}{dx} \frac{k}{x} \\ \\ d\int y(x) = \frac{d}{dx} \frac{k}{x} \\ \\ y(x) = k \frac{d}{dx}\left ( \frac{1}{x} \right ) \\ \\ y(x) = -\frac{k}{x^2}$$

I'm not sure anyone would buy your reasoning, especially dividing by dx that you have done.
For problem 3, ∫y(x)dx = kx, differentiate both sides with respect to x to get y(x) = k.
For problem 4, ∫y(x)dx = k/x, differentiate both sides with respect to x to get y(x) = -k/x².
For problem 1, which you didn't try, ∫y(x)dx = ky. As before, differentiate both sides with respect to x to get y(x) = k dy/dx. This is a separable differential equation that is pretty easy to solve.