Solve the separable differential equation 11x-6ysqrt(x^2-1)dy/dx=0.with the initial condition y(0)=8
11x - 6y√(x^2-1) dy/dx=0
6y dy = 11x/√(x^2-1) dx
3y^2 = 11√(x^2-1) + C
The initial condition is bogus, since y(0) is undefined.
To solve the separable differential equation 11x - 6y√(x^2-1) dy/dx = 0, we need to separate the variables and then integrate.
1. Rearrange the equation by dividing both sides by (11x - 6y√(x^2-1)):
dy/dx = 0 / (11x - 6y√(x^2-1))
dy/dx = 0
2. Integrate both sides with respect to x:
∫ dy = ∫ 0 dx
y = C (where C is the constant of integration)
3. Use the initial condition y(0) = 8 to find the value of C:
8 = C
Therefore, the solution to the given separable differential equation is:
y = 8
To solve the separable differential equation, we'll separate the variables by grouping the terms with "x" on one side and the terms with "y" on the other side of the equation. Then, we can integrate both sides to find the general solution.
The given differential equation is:
(11x - 6ysqrt(x^2 - 1)) dy/dx = 0
Let's split the variables:
(11x) dx = (6ysqrt(x^2 - 1)) dy
Now, we can integrate both sides of the equation:
∫(11x) dx = ∫(6ysqrt(x^2 - 1)) dy
Integrating the left side with respect to "x" gives us:
(11/2)x^2 + C1
Integrating the right side with respect to "y" requires a substitution. Let's substitute u = x^2 - 1, then du = 2x dx:
∫(6y√u) dy = ∫(3y(2u^(1/2))/2) dy
= 3/2 ∫(yu^(1/2)) dy
= 3/2 ∫(yu^(1/2)) dy
= 3/4 ∫(2yu^(1/2)) dy
= 3/4 ∫(2y(x^2 - 1)^(1/2)) dy
Notice that (x^2 - 1)^(1/2) can be written as (√u), so the integral becomes:
3/4 ∫(2y√u) dy
Now integrating this with respect to "y" gives us:
(3/4)(2/3)y^(3/2)√u + C2
= (y^(3/2)√u/2) + C2
Since we have "y" and "x" in our solution, we need to substitute u back in terms of x:
u = x^2 - 1
x^2 = u + 1
x = ±√(u + 1)
Now we can substitute everything back into the equation to find the final solution:
(y^(3/2)√u/2) + C2 = (11/2)x^2 + C1
Substituting u = x^2 - 1:
(y^(3/2)√(x^2 - 1)/2) + C2 = (11/2)x^2 + C1
Finally, we can use the initial condition y(0) = 8 to find the particular solution. Plugging in x = 0 and y = 8:
(8^(3/2)√(0^2 - 1)/2) + C2 = (11/2)(0)^2 + C1
(8^(3/2)(-1)/2) + C2 = 0 + C1
-8√2 + C2 = C1
Therefore, the particular solution is:
(y^(3/2)√(x^2 - 1)/2) = (11/2)x^2 - 8√2 + C2
This is the general solution to the given differential equation with the initial condition y(0) = 8.