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.
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))?
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