"Solve differential equation(x2−y2+2x−y)dx+(x2−y2+x−2y)dy=0" was answered by our community

Poemslore8ye

Answered question

2022-04-20

Solve differential equation

(x^{2} - y^{2} + 2 x - y) d x + (x^{2} - y^{2} + x - 2 y) d y = 0

Answer & Explanation

Addison Zamora

Beginner2022-04-21Added 10 answers

Step 1
Hint: make a substitution

u = x + y, v = x - y, u v = x^{2} - y^{2}, u + v = 2 x, u - v = 2 y

The new equation is:

(u v + u + v - 0.5 u + 0.5 v) (d u + d v) + (u v + 0.5 u + 0.5 v - u + v) (d u - d v) = 0

which simplifies to

v (2 u + 3) d u + u d v = 0

.
This is a separable equation.

4dorts3x

Beginner2022-04-22Added 9 answers

Step 1
Let
$x + y = z$
$x - y = \tilde{z}$
Using this substitution, the differential equation you posted can be written as
$(z \tilde{z} + \tilde{z}) d z$
$= - \frac{1}{2} (z d \tilde{z} + \tilde{z} d z)$
By just adding
$- x d x + y d y = - \frac{1}{2} d (x^{2} - y^{2})$
on both sides. Now, this resulting equation can be easily solved by simplifying it as
$\frac{2}{z} (z + \frac{3}{2}) d z = - \frac{d \tilde{z}}{\tilde{z}}$
Giving
$e^{2 z} z^{3} \tilde{z} = k$
Where k is the integration constant.
Or,
The solution to the differential equation you posed is the solution to equation
$e^{2 (x + y)} {(x + y)}^{3} (x - y) = k$

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