Question

4y+6= (2y+3) дробь dy dx

Answer 1

Відповідь:

Для решения данного дифференциального уравнения нужно привести его к виду, который можно решить с помощью метода разделения переменных. Сначала раскроем скобки справа:

4y + 6 = 2y(dy/dx) + 3(dy/dx)

Затем сгруппируем переменные:

(2y + 3)(dy/dx) = 4y + 6

Далее разделим обе части на (2y + 3):

(dy/dx) = (4y + 6)/(2y + 3)

(dy/dx) = 2(2y + 3)/(2y + 3)

(dy/dx) = 2

Теперь проинтегрируем обе части по переменной y:

∫ dy = ∫ 2 dx

y = 2x + C

где C - произвольная постоянная интегрирования.

Таким образом, общее решение дифференциального уравнения имеет вид:

y = 2x + C

где C - произвольная постоянная.

Покрокове пояснення:

Answer 2

To solve the differential equation $4y + 6 = \frac{dy}{dx}(2y + 3)$, you'll need to separate the variables and then integrate.

First, let's isolate the variables. Rearrange the equation to have all the $y$ terms on one side and the $x$ terms on the other side:

$4y + 6 = \frac{dy}{dx}(2y + 3)$

Now, divide both sides by $(2y + 3)$:

$\frac{4y + 6}{2y + 3} = \frac{dy}{dx}$

Next, let's separate the variables by multiplying both sides by $dx$ and dividing both sides by $(4y + 6)$:

$\frac{1}{4y + 6}dy = dx$

Now, you have separated the variables. To solve this differential equation, you can integrate both sides:

$\int \frac{1}{4y + 6}dy = \int dx$

The left-hand side can be simplified by factoring out a constant from the denominator:

$\frac{1}{4}\int \frac{1}{y + \frac{3}{2}}dy = \int dx$

Now, you can perform the integrations:

$\frac{1}{4}\ln|y + \frac{3}{2}| = x + C$

Here, $C$ is the constant of integration.

To solve for $y$, you'll need to eliminate the natural logarithm by taking the exponent of both sides:

$\ln|y + \frac{3}{2}| = 4x + 4C$

Now, you can remove the absolute value by taking the exponential of both sides:

$|y + \frac{3}{2}| = e^{4x + 4C}$

Since $e^{4x + 4C}$ is always positive, you can drop the absolute value:

$y + \frac{3}{2} = e^{4x + 4C}$

Finally, you can isolate $y$:

$y = e^{4x + 4C} - \frac{3}{2}$

You can combine $e^{4x + 4C}$ into a single constant, so the final solution is:

$y = Ae^{4x} - \frac{3}{2}$

Here, $A$ is a constant that combines $e^{4C}$.