Question

Dy/dx-2y(x+1)=(x+1)^2

Answer 1

РЕШИТЬ ДАННУЮ МАТЕМАТИЧЕСКУЮ ЗАДАЧУ НЕВОЗМОЖНО

УСЛОВИЕ НЕ ПОЛНОЕ!!!

Answer 2

To solve the given first-order linear differential equation, which is in the form of a linear ordinary differential equation (ODE):

dy/dx - 2y(x+1) = (x+1)^2

We can use the method of integrating factors. Here's how you can solve it step by step:

Step 1: Write the equation in standard form dy/dx - 2y(x+1) = (x+1)^2

Step 2: Identify the integrating factor (IF) The integrating factor (IF) is given by the exponential of the integral of the coefficient of 'y' (which is -2(x+1)) with respect to 'x':

IF = e^(-2∫(x+1) dx) = e^(-2(x^2/2 + x) + C1) [integrating with respect to x] = e^(-x^2 - 2x + C1)

Step 3: Multiply both sides of the differential equation by the integrating factor e^(-x^2 - 2x + C1) [dy/dx - 2y(x+1)] = e^(-x^2 - 2x + C1) (x+1)^2

Step 4: Simplify and integrate both sides Now, on the left-hand side, you have the derivative of the product of the integrating factor and y, which you can integrate with respect to 'x':

∫[e^(-x^2 - 2x + C1) (dy/dx - 2y(x+1))] dx = ∫[e^(-x^2 - 2x + C1) (x+1)^2] dx

This will involve integrating by parts and solving for the integral of (x+1)^2 on the right-hand side.

Step 5: Solve for 'y' After integrating both sides, you'll end up with an expression involving 'y' on one side. Solve for 'y' to obtain the solution of the differential equation.

Please note that the calculations involved in this process can be quite intricate and may require some careful algebraic manipulation. If you're working on a specific problem, I recommend going through these steps with a pen and paper or using a computer algebra system to handle the calculations.