Question

Solve dy/dx = (1-x)(1-y)

Answer 1

`y=1-e^{c_(2)}}*e^(-x) *e^(x^2)`

Explanation:

We begin with the differential equation
`(dy)/(dx) =(1-x)(1-y)`

Firstly, we need to get the
`y` and
`dy` as well as the
`x` and
`dx` on the same sides as each other

To do this, we can multiply each side by
`dx` and divide each side by
`(1-y)`

Doing this will give us the following differential

`(1)/(1-y) dy=(1-x)dx`

Now, we can integrate each side

`\int\limits(1)/(1-y) \, dy =\int (1-x) \, dx\\\\-ln(1-y)=x-x^2+c_(1)`

Now, we need to solve for y

`-ln(1-y)=x-x^2+c_(1)\\\\ln(1-y)=x^2-x+c_(2) \\\\1-y=e^{x^2-x+c_(2)} \\\\y=1-e^{x^2-x+c_(2)}\\\\y=1-e^{c_(2)}}*e^(-x) *e^(x^2)`