xdy-ydx=sqrt(x^2-y^2)dx
wolframalpha shows the steps for arriving at the solution
y = x sinh(ln(x)+c)
The given equation is a first-order nonlinear ordinary differential equation. To solve it, we can use a technique called separation of variables.
Step 1: Rewrite the equation in a suitable form.
xdy - ydx = sqrt(x^2 - y^2)dx
Step 2: Rearrange the equation to separate the variables.
xdy - sqrt(x^2 - y^2)dx = ydx
Step 3: Divide both sides of the equation by x.
(dy/dx) - sqrt(x^2 - y^2)/x = y/x
Step 4: Group the terms involving y on one side.
(dy/dx) = (sqrt(x^2 - y^2)/x) + (y/x)
Step 5: Multiply both sides of the equation by dx.
dy = (sqrt(x^2 - y^2)/x)dx + (y/x)dx
Step 6: Separate the variables.
dy/(sqrt(x^2 - y^2) + y) = dx/x
Step 7: Integrate both sides.
∫(1/(sqrt(x^2 - y^2) + y))dy = ∫(1/x)dx
To evaluate these integrals, we can make a suitable substitution.
Step 8: Let z = x^2 - y^2.
dz = (2x - 2yy')dx
dx = dz / (2x - 2yy')
Substituting this into the first integral:
∫(1/(sqrt(x^2 - y^2) + y))dy = ∫(1/(sqrt(z) + y))dy
After integrating this expression, substitute back z = x^2 - y^2.
Similarly, substitute dx = dz / (2x - 2yy') into the second integral:
∫(1/x)dx = ∫(1/(2x - 2yy'))dz
After evaluating this integral, substitute back z = x^2 - y^2.
Step 9: Solve the resulting integrals and substitute back into the original variables to find the solution y(x).
The above steps involve a series of substitutions and integrations, and it is quite involved to perform them fully without knowing the specific values of x and y. So, it may be more suitable to utilize numerical methods or advanced mathematical software to approximate the solution to this differential equation.