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

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Solve dy/dx = (1-x)(1-y) - id116702538, dcardenas2005

jaredfifa

28.01.2022

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

Step-by-step explanation:

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

Firstly, we need to get the and as well as the and on the same sides as each other

To do this, we can multiply each side by 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

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

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