How do we integrate the following with respect to x? Note that y is a function of x

(dy/dx)+(((y^2) - c)^(1/2))?

1 answer

This is a first order nonlinear differential equation because of the square root term. There is no standard method for solving all nonlinear differential equations, but here are the steps for this one:

1. Separate the variables:
Start with (dy/dx)+(((y^2) - c)^(1/2)) = 0.
Rearrange to get (dy/dx) = -(((y^2) - c)^(1/2)).
Multiply both sides by (dy/((y^2) - c)^(1/2)) to get:
(dy/((y^2) - c)^(1/2)) = -dx.

2. Integrate both sides:
Integrating the left side requires a substitution. Let u = y^2 -c. Then du/dy = 2y, so dy = (du/2y). Substituting, we get:
∫(1/u)^(1/2) du = -∫dx.
This simplifies to:
2(u)^(1/2) = -x + C,
where C is the constant of integration.

3. Substitute back for u:
We had u = y^2 -c, so:
2(y^2 -c)^(1/2) = -x + C.

This is the general solution to the differential equation. It cannot be expressed in terms of elementary functions, so the best we can do is solve for y explicitly in some cases. For example, if we know that y(0) = a, we can solve for C and get:
C = 2(a^2 -c)^(1/2).
Substituting this into the general solution and solving for y, we get:
y = ±[(x - 2(a^2 -c)^(1/2))/2]^2 + c.
Note that there are two solutions, one with the plus sign and one with the minus sign. The choice between them depends on the initial condition or other physical considerations.