Question

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

Answer 1

Answer:

$y=1-e^{c_{2}}}*e^{-x} *e^{x^2}$

Step-by-step explanation:

We begin with the differential equation $\frac{dy}{dx} =(1-x)(1-y)$

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

$\frac{1}{1-y} dy=(1-x)dx$

Now, we can integrate each side

$\int\limits\frac{1}{1-y} \, dy =\int (1-x) \, dx\\\\-ln(1-y)=x-x^2+c_{1}$

Now, we need to solve for y

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