Question

уравнение решить, методом бернулли) y'=y/x+1/y спасибо)

Answer 1

решение в файле. Уравнение приведено к виду, которое решается методом Бернулли

Answer 2

Solving the Differential Equation using the Bernoulli Method

To solve the differential equation y' = y/x + 1/y using the Bernoulli method, we can follow these steps:

Step 1: Identify the form of the equation The given equation is in the form y' = P(x)y + Q(x)y^n, where n ≠ 1. In this case, P(x) = 1/x and Q(x) = 1.

Step 2: Make the substitution Let's make the substitution v = y^(1-n). In this case, n = -1, so v = y^2.

Step 3: Differentiate both sides of the equation Differentiating v = y^2 with respect to x gives us dv/dx = 2yy'.

Step 4: Substitute the expressions into the differential equation Substituting v = y^2 and dv/dx = 2yy' into the original differential equation y' = y/x + 1/y, we get 2yy' = y^2/x + 1/y.

Step 5: Simplify the equation Multiplying both sides of the equation by x gives us 2xyy' = y^2 + x/y.

Step 6: Solve the resulting linear differential equation The resulting equation 2xyy' = y^2 + x/y is a linear differential equation. We can solve it using standard methods.

Unfortunately, I couldn't find a specific solution to this equation using the Bernoulli method in the available search results. It's possible that the equation may not have an elementary solution or that a different method may be more appropriate for solving it.

If you have any further questions or need clarification, please let me know!