how do i solveseparable differential equation

7x−4y sqrt(x^2+1) dxdy=0
Subject to the initial condition: y(0)=−10

Is that dx/dy or dy/dx in your equation?

If it is dx/dy, rewrite it as
7x − 4y sqrt(x^2+1) dx/dy = 0 as
7x = 4y sqrt(x^2+1) dx/dy
which can be separated as
(1/7x)*sqrt(x^2+1) dx = dy/4y

Now integrate both sides separately.

To solve the separable differential equation 7x - 4y√(x^2 + 1) dxdy = 0 with the initial condition y(0) = -10, you can follow these steps:

Step 1: Rewrite the equation in the form of dy/dx = f(x)g(y)
Divide both sides of the equation by (7x - 4y√(x^2 + 1)):
(dy/dx) / (7x-4y√(x^2+1)) = 0

Step 2: Separate variables
Multiply both sides by dx:
dy / (7x-4y√(x^2 + 1)) = 0 dx

Step 3: Integrate both sides of the equation
Integrate the left side with respect to y and the right side with respect to x:
∫ [1 / (7x-4y√(x^2+1))] dy = ∫ 0 dx

Step 4: Solve the integrals
To simplify the integration, let u = x^2 + 1, then du = 2xdx:
∫ [1 / (7x-4y√(x^2+1))] dy = ∫ [1 / (7x-4y√u)] dy
Using the substitution, the integral becomes:
∫ [1 / (7x-4y√u)] dy = ∫ [1 / (7x-4y√u)] (du / (2x))

Simplify the expression:
∫ [1 / (7x-4y√u)] dy = (1/2) ∫ [(1 / (7-4(√u / x))) ] du

Let v = √u / x, then dv = (1/x) * (1/2√u) du:
∫ [(1 / (7-4(√u / x)))] du = ∫ [(1/(7-4v)) * 2v dv]

Rewrite the expression in terms of v:
∫ [(1/(7-4v)) * 2v dv] = 2 ∫ [(v/(7-4v))] dv

Integrate the expression:
2 ∫ [(v/(7-4v))] dv = -2 ln |7-4v| + C

Here C is the constant of integration.

Step 5: Evaluate the definite integral
Now that we have the integral equation, evaluate the definite integral for v from y(0) = -10 to y:
-2 ∫ ln |7-4v| dv = -2 [ln |7-4v|] |_(-10)^(y)]

Evaluate the definite integral and simplify the expression:
-2 [ln |7-4v|] |_(-10)^(y)] = -2 [ln |7-4y| - ln |7+40|] = -2 ln |(7-4y)/(47)|

Step 6: Solve for y
Set the result of the definite integral equal to a constant A:
-2 ln |(7-4y)/(47)| = A

Solve for y:
ln |(7-4y)/(47)| = -A/2

Take the exponential of both sides:
|(7-4y)/(47)| = e^(-A/2)

Consider the absolute value:
(7-4y)/(47) = ± e^(-A/2)

Solve for y:
y = (7 ± 47e^(-A/2)) / 4

Note: The constant A can be determined using the initial condition y(0) = -10.

So, the solution to the separable differential equation with the given initial condition is y = (7 ± 47e^(-A/2)) / 4, where A is determined based on the initial condition.

To solve the given separable differential equation:

7x - 4y√(x^2 + 1) dxdy = 0

Subject to the initial condition: y(0) = -10

Step 1: Separate the variables. Move all terms involving 'x' to one side and all terms involving 'y' to the other side:

7x dxdy = 4y√(x^2 + 1) dxdy

Step 2: Divide both sides by 'x' and multiply by 'dy' to isolate the terms involving 'y' on one side and the terms involving 'x' on the other side:

7 dxdy/x = 4y/√(x^2 + 1) dxdy

Step 3: Now integrate both sides with respect to their respective variables.

∫ 7 dxdy/x = ∫ 4y/√(x^2 + 1) dxdy

On the left side, the integral with respect to 'x' should be evaluated first since we are integrating with respect to 'x' on that side. Similarly, on the right side, the integral with respect to 'y' should be evaluated first since we are integrating with respect to 'y' on that side.

Step 4: Integrate the left side with respect to 'x':

∫ 7 dxdy/x = 7∫ dx/x = 7ln|x|

Step 5: Integrate the right side with respect to 'y':

∫ 4y/√(x^2 + 1) dxdy

For this integral, we can use a substitution. Let u = x^2 + 1. Then du/dx = 2x. Rearranging this equation, we get dx = du/(2x). Substituting into the integral:

∫ 4y/√(x^2 + 1) dxdy = 4∫ y/√u (du/2x) dy

= 2∫ y/√u dy

Now, integrate the right side with respect to 'y':

= 2∫ y/√u dy = 2∫ y/√u dy = 2 * (2/3) * u^(3/2) + C

Now substitute back u = x^2 + 1:

= 2 * (2/3) * (x^2 + 1)^(3/2) + C

Step 6: Set up the complete solution by combining both sides:

7ln|x| = 2 * (2/3) * (x^2 + 1)^(3/2) + C

Step 7: Apply the initial condition to find the value of the constant 'C'.

When x = 0, y = -10. Substitute these values into the equation:

7ln|0| = 2 * (2/3) * (0^2 + 1)^(3/2) + C
Since the natural logarithm of zero is undefined, we can conclude that x ≠ 0.

Therefore, there is no solution that satisfies the initial condition y(0) = -10.