1) What are the equilibrium solutions to the differential equation and determine if it is stable or unstable with the initial condition y(-4)=1:

0.1(y+2)(4-y)
2) Use Euler's method with step size=0.5 and initial condition y(0)=3 to solve the equation y(0)=3 to estimate y(2), Then solve the equation with the initial condition y(0)=3 and step size 0.1.
Then solve to find the actual value of y(2) and compare.
3) Solve the differential equation for initial condition y(0)=3 for dy/dt=t/y.
Explain why dy/dx=(x+y)/(x-y) is not a separable differential equation.

1) To find the equilibrium solutions of a differential equation, you set the equation equal to zero and solve for y. In this case, the given differential equation is 0.1(y+2)(4-y).

Step 1: Set the equation equal to zero:
0.1(y+2)(4-y) = 0

Step 2: Solve for y by factoring or using the zero-product property:
(y+2)(4-y) = 0

To solve for y, we have two possibilities:
y+2 = 0 OR 4-y = 0

For the first equation, y = -2.
For the second equation, y = 4.

So the equilibrium solutions for the given differential equation are y = -2 and y = 4.

To determine if these equilibrium solutions are stable or unstable, you need to analyze the behavior of the differential equation around these points. One way to do this is by using the concept of a sign chart or by considering the sign of the derivative around these points.

2) Euler's method is a numerical method used to approximate the solution of a differential equation. Given a step size, initial condition, and an equation, we can estimate the value of y at a specific point.

a) To use Euler's method with a step size of 0.5 and an initial condition of y(0) = 3, we can follow these steps:

- Start with the initial condition: y(0) = 3.
- Calculate the slope of the curve at that point: dy/dx = (x + y) / (x - y).
- Use the slope and step size to find the change in y: Δy = slope * step size.
- Update the y-value: y(0.5) = y(0) + Δy.
- Repeat the previous steps until you reach the desired point.
- For y(2), the process will be repeated until we reach x = 2.

b) Repeat the previous steps using a step size of 0.1 instead:
- Start with the initial condition: y(0) = 3.
- Calculate the slope of the curve at that point: dy/dx = (x + y) / (x - y).
- Use the slope and step size to find the change in y: Δy = slope * step size.
- Update the y-value: y(0.1) = y(0) + Δy.
- Repeat the previous steps until you reach the desired point.
- For y(2), the process will be repeated until we reach x = 2.

After finding the values of y(2) using both step sizes, you can compare the results with the actual value of y(2) to evaluate the accuracy of Euler's method.

3) The given differential equation is dy/dt = t/y with the initial condition y(0) = 3.

To solve this equation, we will employ the method of separation of variables:

Step 1: Rewrite the equation to separate the variables:
dy/y = t dt

Step 2: Integrate both sides with respect to their respective variables:
∫ dy/y = ∫ t dt

Step 3: Evaluate the integrals on both sides:
ln|y| = (1/2)t^2 + C

Step 4: Solve for y by exponentiating both sides:
|y| = e^((1/2)t^2 + C)

Step 5: Consider the absolute value for the general solution:
y = ±e^C * e^((1/2)t^2)

Step 6: Apply the initial condition y(0) = 3 to find the value of the constant C:
3 = ±e^C * e^(0)
3 = ±e^C

Since e^C is always positive, the equation becomes:
3 = e^C

Taking the natural logarithm of both sides:
ln(3) = C

Step 7: Substitute the value of C back into the general solution:
y = ±e^(ln(3)) * e^((1/2)t^2)
y = ±3 * e^((1/2)t^2)

Therefore, the solution to the given differential equation with the initial condition y(0) = 3 is y = ±3 * e^((1/2)t^2).

4) The differential equation dy/dx = (x+y)/(x-y) is not a separable differential equation because the variables x and y are not separated on opposite sides of the equation.

In a separable differential equation, you can rewrite the equation so that all x terms are on one side and all y terms are on the other side. This allows you to integrate each side separately.

However, in the given equation dy/dx = (x+y)/(x-y), both x and y appear in the numerator and denominator on the right side of the equation. Therefore, it is not possible to separate the variables x and y and integrate them independently.

To solve this type of equation, alternative methods such as integrating factors, exact equations, or numerical methods may be required.