Question

Y'' = √x-sin 2x x0=1 y(0)=-1/8 y'(0)=1/8cos 2 y'(0)=1/2

Answer 1

Ответ:

y'' = √x-sin 2x x0=1 y(0)=-1/8 y'(0)=1/8cos 2 y'(0)=1/2y'' = √x-sin 2x x0=1 y(0)=-1/8 y'(0)=1/8cos 2 y'(0)=1/2

Пошаговое объяснение:

Izichka

Answer 2

It appears that there might be a typo in the information you provided. You've mentioned two values for y'(0), which is not consistent. Please verify the correct initial conditions for y'(0) so that I can assist you with the given differential equation.

In the meantime, I can help you solve the differential equation Y'' = √x - sin(2x) with the initial conditions x₀ = 1 and y(0) = -1/8.

To solve this differential equation, we'll follow these steps:

Step 1: Integrate the given equation to obtain y'. Step 2: Solve the equation obtained in step 1 to find y. Step 3: Apply the initial conditions to determine the specific solution.

Let's begin:

Step 1: Integrate the given equation to obtain y' Integrating both sides of the differential equation Y'' = √x - sin(2x) with respect to x once, we get: y' = ∫ (√x - sin(2x)) dx

To integrate the terms on the right-hand side, we split the integral as follows: y' = ∫ (√x dx) - ∫ (sin(2x) dx)

Integrating each term separately: ∫ (√x dx) = (2/3) * x^(3/2) ∫ (sin(2x) dx) = -(1/2) * cos(2x)

Combining the integrals: y' = (2/3) * x^(3/2) - (1/2) * cos(2x) + C

Here, C represents the constant of integration.

Step 2: Solve the equation obtained in step 1 to find y To find y, we integrate y' with respect to x: y = ∫ [(2/3) * x^(3/2) - (1/2) * cos(2x) + C] dx

Integrating each term separately: ∫ [(2/3) * x^(3/2) - (1/2) * cos(2x) + C] dx = (2/5) * x^(5/2) - (1/4) * sin(2x) + Cx + D

Here, D is the constant of integration.

Step 3: Apply the initial conditions to determine the specific solution Using the initial condition y(0) = -1/8, we substitute x = 0 and y = -1/8 into the equation obtained in step 2: (-1/8) = (2/5) * (0)^(5/2) - (1/4) * sin(2(0)) + C * 0 + D (-1/8) = 0 - 0 + D D = -1/8

Therefore, the specific solution for the given initial conditions is: y = (2/5) * x^(5/2) - (1/4) * sin(2x) - (1/8) * x - 1/8