I am having alot of trouble remembering how to do this and this is a practice question for my final if you could help that would be great. Solve the differential equation (dy/dx)=(3x)/(1-x^2)
dy = (3x)/(1-x^2) dx
y = Integral of [3x/(1-x^2)] dx + constant
For the integration, let
1-x^2 = u
-2x dx = du
Which make the integral
Integral of (-3/2)du/u = -(3/2)ln(1-x^2)
Sure, I can help you solve this differential equation.
To solve the differential equation (dy/dx) = (3x)/(1-x^2), we can use the method of separation of variables.
Step 1: Write the given differential equation in the form of dy = f(x) dx, where f(x) is a function of x only. In this case, we have:
dy = (3x)/(1-x^2) dx
Step 2: Separate the variables by moving all the x terms to the right side and all the y terms to the left side. In this case, we have:
(1-x^2) dy = 3x dx
Step 3: Integrate both sides of the equation with respect to their respective variables. The integral of the left side will give us the antiderivative function of dy, while the integral of the right side will give us the antiderivative function of dx.
∫ (1-x^2) dy = ∫ 3x dx
Step 4: Evaluate the integrals.
For the left side, we can expand (1-x^2) as (1- x^2) = (1 + x)(1 - x):
∫ (1 + x)(1 - x) dy = ∫ 3x dx
∫ (1 + x - x - x^2) dy = ∫ 3x dx
∫ (1 - x^2) dy = ∫ 3x dx
Now, integrate both sides:
y - (1/3)x^3 + C1 = (3/2)x^2 + C2
Where C1 and C2 are arbitrary constants of integration.
This equation represents the general solution to the given differential equation.
Step 5: If you have any initial conditions given, you can use them to solve for the values of the constants C1 and C2. Substitute the values of x and y from the initial condition into the equation and solve for the constants.
However, since you haven't provided any initial conditions, we have obtained the general solution to the differential equation.