Question

Помогите пожалуйста решить: 1) dy/dx=y+x 2) интеграл 3sin5xdx За ранее спасибо)

Answer 1

1) Ответ: y*x-y-x=0
2) Ответ: 3*sin(5)*x^2*d

Answer 2

Solving the Differential Equation

To solve the differential equation dy/dx = y + x, we can use the method of integrating factors. First, we'll rewrite the equation in the form dy/dx - y = x. Then, we'll find the integrating factor, which is given by the exponential of the integral of the coefficient of y with respect to x.

The integrating factor (IF) is given by: IF = e^∫(-1)dx = e^(-x)

Multiplying both sides of the differential equation by the integrating factor gives: e^(-x)dy/dx - e^(-x)y = xe^(-x)

This can be rewritten as: d/dx (e^(-x)y) = xe^(-x)

Integrating both sides with respect to x gives: e^(-x)y = -xe^(-x) - e^(-x) + C

Where C is the constant of integration. Solving for y gives: y = -x - 1 + Ce^(x)

This is the general solution to the differential equation.

Evaluating the Integral

To evaluate the integral of 3sin(5x)dx, we can use the reverse chain rule, also known as u-substitution. Let's let u = 5x, then du = 5dx. This gives us:

∫3sin(5x)dx = (3/5)∫sin(u)du = -(3/5)cos(u) + C = -(3/5)cos(5x) + C

Where C is the constant of integration.

So, the result of the integral is -(3/5)cos(5x) + C.