solve the initial value problem of the differential equation (y+sqrt(x^2+y^2))dx-xdy=0,y(3)=4

This is another homogeneous equation. Substitute y = vx and you end up with

dv/√(1+v^2) = -1/x dx

To solve the initial value problem of the given differential equation, we will follow these steps:

Step 1: Rewrite the equation in a more convenient form.
Step 2: Determine if the equation is exact or not.
Step 3: Solve the equation for y(x).
Step 4: Substitute the initial condition to find the constant of integration.
Step 5: Write the final solution to the initial value problem.

Let's go through each step in detail:

Step 1: Rewrite the equation in a more convenient form.
The given differential equation is (y + sqrt(x^2 + y^2))dx - xdy = 0. We will rearrange it to separate the variables:
(y + sqrt(x^2 + y^2))dx = xdy

Step 2: Determine if the equation is exact or not.
To determine if the equation is exact, we check if the partial derivative with respect to y of the coefficient of dx (i.e., y + sqrt(x^2 + y^2)) is equal to the partial derivative with respect to x of the coefficient of dy (i.e., -x). Let's calculate these partial derivatives:

∂/∂y(y + sqrt(x^2 + y^2)) = 1 + (1/2) * (2y) / sqrt(x^2 + y^2)
∂/∂x(-x) = -1

Since the partial derivatives are not equal, the equation is not exact.

Step 3: Solve the equation for y(x).
To solve this type of equation, we use an integrating factor to make it exact. The integrating factor is the exponential of the integral of the term multiplying dx (i.e., (1 + (1/2) * (2y) / sqrt(x^2 + y^2))):

Integrating factor (IF) = exp ∫ (1 + (1/2) * (2y) / sqrt(x^2 + y^2)) dx
= exp ∫ (1 + y / sqrt(x^2 + y^2)) dx
= exp(x + sqrt(x^2 + y^2))

Now we multiply the entire equation by the integrating factor:

exp(x + sqrt(x^2 + y^2)) * (y + sqrt(x^2 + y^2)) dx = exp(x + sqrt(x^2 + y^2)) * x dy

Expanding the left side and simplifying:

y * exp(x + sqrt(x^2 + y^2)) dx + sqrt(x^2 + y^2) * exp(x + sqrt(x^2 + y^2)) dx = x * exp(x + sqrt(x^2 + y^2)) dy

Step 4: Substitute the initial condition to find the constant of integration.
Substituting the initial condition y(3) = 4 into the equation:

4 * exp(3 + sqrt(3^2 + 4^2)) * d3 + sqrt(3^2 + 4^2) * exp(3 + sqrt(3^2 + 4^2)) * d3 = 3 * exp(3 + sqrt(3^2 + 4^2)) * d4

Simplifying, we get:

4 * exp(3 + 5) * d3 + 5 * exp(3 + 5) * d3 = 3 * exp(3 + 5) * d4

4 * exp(8) * d3 + 5 * exp(8) * d3 = 3 * exp(8) * d4

9 * exp(8) * d3 = 3 * exp(8) * d4

Dividing both sides by 3 * exp(8):

3 * d3 = d4

Step 5: Write the final solution to the initial value problem.
Integrating both sides with respect to their respective variables:

∫3 * d3 = ∫1 * d4

3 * 3 = 4 + C

9 = 4 + C

C = 9 - 4

C = 5

Therefore, the constant of integration is 5.

The final solution to the initial value problem is:

y = 3x - 1 - sqrt(x^2 + (3x-1)^2)