The differential equation x · y' + y² = 0 in differential form is x · dy + y² · dx = 0.
The standard form of the differential equation (y')² - 5 · (y') + 6 = (x + y) · (y' - 2) is the equation y' - y - 3 = x.
In the first part of this problem we have a nonlinear diferential equation of the form f(x, y, y') = 0, which has to be changed into diferential form, defined below:
M(x, y) dx + N(x, y) dy = 0, where y' = dy / dx. (1)
Now we proceed to work on the ordinary differential equation:
x · y' + y² = 0
x · y' = - y²
x · (dy / dx) = - y²
x · dy = - y² · dx
x · dy + y² · dx = 0
The differential equation x · y' + y² = 0 in differential form is x · dy + y² · dx = 0.
The second part of the problem involves another nonlinear differential equation, of which we must find its standard form, also defined below:
f(y', y) = g(x) (2)
Finally, we proceed to modify the equation:
(y')² - 5 · (y') + 6 = (x + y) · (y' - 2)
(y' - 3) · (y' - 2) = (x + y) · (y' - 2)
y' - 3 = x + y
y' - y - 3 = x
The standard form of the differential equation (y')² - 5 · (y') + 6 = (x + y) · (y' - 2) is the equation y' - y - 3 = x.
The statement presents typing mistakes, correct form is shown below:
3. Write the differential equation x · y' + y² = 0 in differential form. Write the differential equation (y')² - 5 · y' + 6 = (x + y) · (y' - 2).
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