Solve the differential equation dy/dx=3x^2y^2 with the condition that y(1)=4.
I know that y= -1/(x^3 + c) but what do I do with the y(1)=4 part?
Oh differential equations, the mathematician's version of a roller coaster ride! Don't worry, I'm here to make it a little more fun for you.
First, let's tackle the differential equation dy/dx = 3x^2y^2. Now, this equation is separable, which means we can separate the variables and integrate. So, let's do some magic!
Rearranging the equation, we'll get dy/y^2 = 3x^2 dx. Now, let's integrate both sides:
∫(1/y^2) dy = ∫3x^2 dx
Integrating the left side will give us ∫(1/y^2) dy = -1/y + C1, where C1 is the constant of integration. On the right side, we have ∫3x^2 dx = x^3 + C2, where C2 is another constant of integration.
So, our equation becomes -1/y + C1 = x^3 + C2. Rearranging it a bit, we get -1/y = x^3 + C - C1, where C = C2 - C1. Finally, let me sprinkle some clown magic on this equation:
-1/y = x^3 + C - C1
Now, let's solve for y. Multiply both sides by -1:
1/y = -(x^3 + C - C1)
Take the reciprocal of both sides:
y = -1/(x^3 + C - C1)
Now, to find the value of C, we'll use the condition y(1) = 4. Plugging in x = 1 and y = 4 into our equation, we have:
4 = -1/(1^3 + C - C1)
Simplifying further, we get:
4 = -1/(1 + C - C1)
Cross-multiplying, we have:
4(1 + C - C1) = -1
Simplifying once more, we get:
4 + 4C - 4C1 = -1
4C - 4C1 = -5
Dividing both sides by 4, we have:
C - C1 = -5/4
Now, my friend, you have the task of using this information to solve for C and C1. Let your numerical skills shine! And remember, math can sometimes be strange, but don't let it make you a square!
To solve the differential equation dy/dx = 3x^2y^2 with the condition y(1) = 4, you already have the general solution y = -1/(x^3 + c).
Now, to find the particular solution that satisfies the initial condition y(1) = 4, substitute x = 1 and y = 4 into the general solution:
4 = -1/(1^3 + c)
4 = -1/(1 + c)
-4 = 1 + c
c = -5
Therefore, the particular solution with the initial condition y(1) = 4 is y = -1/(x^3 - 5).
To solve the differential equation dy/dx = 3x^2y^2 with the initial condition y(1) = 4, you can use the method of separation of variables.
Here are the steps to solve the equation:
Step 1: Separate the variables.
dy/y^2 = 3x^2 dx
Step 2: Integrate both sides.
∫(1/y^2) dy = ∫3x^2 dx
On the left side, integrate (1/y^2) dy:
∫(1/y^2) dy = -1/y
On the right side, integrate 3x^2 dx:
∫3x^2 dx = x^3
So the equation becomes:
-1/y = x^3 + C, where C is the constant of integration.
Step 3: Solve for y.
To solve for y, you need to isolate it on one side of the equation. Rearrange the equation to get:
y = -1/(x^3 + C)
Step 4: Use the initial condition.
Now, we'll use the initial condition y(1) = 4 to find the value of the constant, C.
Substitute x = 1 and y = 4 into the equation:
4 = -1/(1^3 + C)
Simplify:
-1/(1 + C) = 1/4
Take the reciprocal of both sides:
1 + C = -4
Solve for C:
C = -5
Step 5: Final solution.
Now that you have the value of the constant C, substitute it back into the equation:
y = -1/(x^3 - 5)
Therefore, the solution to the differential equation dy/dx = 3x^2y^2 with the initial condition y(1) = 4 is y = -1/(x^3 - 5).
you want to find c. So, since y(1) = 4, plug it in
-1/(1^3+c) = 4
-1/4 = 1+c
c = -5/4
Now the particular solution is
y = -1/(x^3 - 5/4)