Question

Xdy+ydx=0y(2)=3y(6)=?​

Answer 1

Перенесём всё в одну часть и домножим на (1/x^2):


Заметим, что в левой части равенства стоит полный дифференциал, а правая часть ноль. Так может быть только если то, что стоит под дифференциалом - константа.


Константу C определим из условия:


Ответ.

Answer 2

This is a differential equation problem. You're given the differential equation:

xdy + ydx = 0

To solve this equation, we can use separation of variables. First, let's rewrite the equation as:

xdy = -ydx

Now, we'll separate the variables by moving the y terms to the left side and the x terms to the right side:

dy/y = -dx/x

Next, we'll integrate both sides:

∫(1/y) dy = ∫(-1/x) dx

ln|y| = -ln|x| + C

Where C is the constant of integration. Now, let's simplify the equation by combining the natural logarithms:

ln|y| + ln|x| = C

Using the properties of logarithms, we can combine the logarithms:

ln(|yx|) = C

Now, we'll exponentiate both sides to get rid of the natural logarithm:

|yx| = e^C

Since C is an arbitrary constant, e^C is also a constant, which we can denote as K:

|yx| = K

Now, let's use the initial conditions provided:

y(2) = 3

|y(2) * 2| = K

3 * 2 = K

K = 6

Now that we have found K, we can use it to find y(6):

|y(6) * 6| = 6

6y(6) = 6

Divide both sides by 6:

y(6) = 1

So, y(6) = 1.