Question

Solve the given initial-value problem. (2y 2t − 9) dt (8y 2t − 1) dy = 0, y(−1) = 2?

Answer 1

The solution to the given initial-value problem is y(t) = 2.

Step-by-step explanation:

The provided initial-value problem is a first-order ordinary differential equation with an initial condition. To solve it, we'll separate variables and integrate. The given equation is:

`\[ (2y²t - 9) dt + (8y²t - 1) dy = 0 \]`

Integrating both sides with respect to their respective variables, we get:

`\[ ∫ (2y²t - 9) dt + ∫ (8y²t - 1) dy = C \]`

Where C is the constant of integration.

Solving the integrals, we obtain:

`\[ y²t² - 9t + 4y²t² - y = C \]`

Simplifying and combining like terms, we have:

`\[ 6y²t² - 9t - y = C \]`

Now, applying the initial condition y(-1) = 2, we can find the value of C. Substituting t = -1 and y = 2 into the equation, we get:

`\[ 6(2)²(-1)² - 9(-1) - 2 = C \]`

Solving this, we find C = 29.

Therefore, the final solution to the initial-value problem is 6y²t² - 9t - y = 29.