solve the initial value problem by seperation of variables du/dt=2t+sec2t/2u, u(0)=-5

I am guessing that you mean sec^2 and left out parentheses

du/dt=(2t+sec^2t)/2u, u(0)=-5

2 u du = 2 t dt + sec^2 t dt

u^2 = t^2 + tan t + c

when t = 0, u = -5

25 = c

u^2 = t^2 + tan t + 25

To solve the initial value problem using separation of variables, we need to separate the variables and then integrate both sides of the equation.

Given the initial value problem: du/dt = 2t + sec^2(t)/2u, with u(0) = -5

Step 1: Separation of variables
Move all the terms involving 'u' to the left side of the equation and all the terms involving 't' to the right side. This can be done by multiplying both sides of the equation by 'dt' and dividing both sides by the expression with 'u'.

(2u)du = (2t + sec^2(t)/2)dt

Step 2: Integrate both sides
Integrate both sides of the equation with respect to 'u' and 't' separately.

∫(2u)du = ∫(2t + sec^2(t)/2)dt

Step 3: Evaluate the integrals
The integral of (2u)du can be directly computed as u^2.

The integral of (2t + sec^2(t)/2)dt can be found using basic integral rules. The integral of 2t with respect to t is t^2. To integrate sec^2(t), use the trigonometric identity: ∫sec^2(t)dt = tan(t). So the integral of sec^2(t)/2 is (1/2)tan(t).

Now, we have:

u^2 = t^2 + (1/2)tan(t) + C ------------------ (1)

where C is the constant of integration.

Step 4: Apply the initial condition
To apply the initial condition u(0) = -5, substitute t=0 and u=-5 into equation (1).

(-5)^2 = (0)^2 + (1/2)tan(0) + C

25 = 0 + 0 + C

C = 25

Step 5: Final solution
Substitute the value of C back into equation (1).

u^2 = t^2 + (1/2)tan(t) + 25

Taking the square root of both sides:

u = ± √(t^2 + (1/2)tan(t) + 25)

Therefore, the solution to the initial value problem is u = ± √(t^2 + (1/2)tan(t) + 25), where u(0) = -5.