Solve the separable differential equation 10x−6ysqrt(x^2+1)dy/dx=0. with the initial condition y(0)=4.
10x−6y√(x^2+1)dy/dx=0
6y√(x^2+1) dy/dx = 10x
6y dy = 10x/√(x^2+1) dx
3y^2 = 10√(x^2+1) + c
Since y(0) = 4,
3*16 = 10+c
c = 38
now you have it.
To solve the separable differential equation 10x - 6ysqrt(x^2+1)dy/dx = 0, we need to separate the variables and integrate.
Step 1: Rewrite the equation in the form dy/dx = f(x)g(y)
10x - 6ysqrt(x^2+1)dy/dx = 0
6ysqrt(x^2+1)dy = 10xdx
Step 2: Separate the variables
(6ysqrt(x^2+1))/10 x dy = dx
Step 3: Integrate both sides
∫(6ysqrt(x^2+1))/10 x dy = ∫dx
(3/5) ∫ysqrt(x^2+1) dy = x + C1
Step 4: Integrate the left side with respect to y
(3/5) ∫ysqrt(x^2+1) dy = (3/5) (∫y(x^2+1)^(1/2) dy)
(3/5) ∫y(x^2+1)^(1/2) dy = (3/5) [(x^2+1)^(3/2) * y / (3/2)] + C2
(3/5) ∫y(x^2+1)^(1/2) dy = (2/5) (x^2+1)^(3/2) * y + C2
Step 5: Incorporate the initial condition y(0) = 4
(3/5) [(0^2+1)^(3/2) * 4 / (3/2)] + C2 = 0 + C1
(3/5) (1)^(3/2) * 4 * 2/3 = C2 - C1
(3/5) * 4 * 2/3 = C2 - C1
(24/15) = C2 - C1
8/5 = C2 - C1
Step 6: Combine the results and solve for y
(2/5) (x^2+1)^(3/2) * y + 8/5 = x + C1
(2/5) (x^2+1)^(3/2) * y = x + C1 - 8/5
(2/5) (x^2+1)^(3/2) * y = (5x + 5C1 - 8)/5
(x^2+1)^(3/2) * y = (5x + 5C1 - 8)(5/2)
(x^2+1)^(3/2) * y = (25x + 25C1 - 40)/2
y = (25x + 25C1 - 40)/(2(x^2+1)^(3/2))
Therefore, the solution to the differential equation with the initial condition y(0) = 4 is:
y = (25x + 25C1 - 40)/(2(x^2+1)^(3/2))
To solve the separable differential equation 10x - 6ysqrt(x^2 + 1)dy/dx = 0 with the initial condition y(0) = 4, we can follow these steps:
Step 1: Separate the variables.
Move the dy/dx term to the left-hand side and the x and y terms to the right-hand side:
6ysqrt(x^2 + 1)dy = 10xdx
Step 2: Integrate both sides.
Integrate both sides of the equation with respect to their respective variables:
∫6ysqrt(x^2 + 1)dy = ∫10xdx
Step 3: Integrate the left-hand side.
To integrate the left-hand side, use the substitution u = x^2 + 1. Then, du/dx = 2x. Rearrange to solve for dx: dx = du/(2x). Substitute this back into the integral:
∫6ysqrt(x^2 + 1)dy = ∫10x(dx) = ∫10x(du/(2x))
Cancel out the x terms in the numerator and denominator:
∫6ysqrt(x^2 + 1)dy = ∫5du
Integrate both sides:
3y(x^2 + 1)^(3/2) = 5u + C1
Step 4: Solve for u.
Substitute back u = x^2 + 1:
3y(x^2 + 1)^(3/2) = 5(x^2 + 1) + C1
Step 5: Use the initial condition to find C1.
Plug in x = 0 and y = 4 into the equation:
12 = 5(0^2 + 1) + C1
12 = 5 + C1
C1 = 7
Step 6: Solve for y.
Substitute C1 back into the equation and solve for y:
3y(x^2 + 1)^(3/2) = 5(x^2 + 1) + 7
Divide both sides by 3(x^2 + 1)^(3/2):
y = (5(x^2 + 1) + 7) / 3(x^2 + 1)^(3/2)
This is the solution to the separable differential equation with the given initial condition.