"Is it possible to solve this first-order O.D.E..." was answered by our experts

Elise Winters

Answered question

2022-04-23

Is it possible to solve this first-order O.D.E using the separable equations method?

\frac{d y}{d x} = x - 1 + x y - y

hadnya1qd

Beginner2022-04-24Added 15 answers

Step 1
You can rewrite the problem as

\frac{d y}{d x} = (x - 1) (y + 1)

and can then use separation of variables (I will write

y^{'}

for

\frac{d y}{d x}

) to get

\frac{y^{'}}{y + 1} = x - 1

Then integrating with respect to x gives

\ln (y + 1) = {\frac{12}{x}}^{2} - x + c

And that gives

y = c e^{\frac{x^{2}}{2} - x} - 1

Celia Morse

Beginner2022-04-25Added 11 answers

Step 1

\frac{d y}{d x} = x + x y - 1 - y = (x - 1) (1 + y)

hence

\frac{y^{'}}{1 + y} = x - 1

\ln (1 + y) = \frac{x^{2}}{2} - x + c

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