Solve dy/dx = y/(2y-x) using rational substitution (using v=y/x) with the initial condition y(0)=1.
I solved this problem and found C (the constant) to be undefined. That is not the correct answer.
(2y -x) dy = y dx
2 y dy - x dy = y dx
2 y dy = x dy + y dx
y^2 = x y + c
when x = 0
y^2 = c = 1
y^2 = x y + 1
To solve the given differential equation using rational substitution, we first substitute y/x with a new variable v. This gives us the equation:
dy/dx = v
Now, let's find dy and dx in terms of v. Rearrange the equation y/x = v to get y = v*x. Differentiating both sides with respect to x, we get dy = v*dx + x*dv.
Substituting these values of dy and dx in the original differential equation, we have:
v*dx + x*dv = v/(2v - 1)
Now we need to separate the variables and integrate both sides. We can rewrite the equation as:
(v*dx)/(v/(2v - 1) - x*dv) = 1
We can simplify the left side by multiplying both numerator and denominator by (2v - 1) to get:
(2v - 1)dx = xdv
Now we separate variables:
(2v - 1)/v dv = dx/x
Integrating both sides gives:
∫(2v - 1)/v dv = ∫dx/x
This leads to:
2∫dv - ∫(1/v) dv = ∫dx
Integrating each term gives:
2v - ln|v| = ln|x| + C,
where C is the constant of integration.
Now, substitute y/x back in for v:
2(y/x) - ln|y/x| = ln|x| + C
Simplifying, we have:
2y - 2x ln|y/x| = x ln|x| + Cx
Finally, substitute the initial condition y(0) = 1:
2(1) - 2(0) ln|1/0| = 0 ln|0| + C(0)
2 = C(0)
Since C is multiplied by 0, it is undefined.
Therefore, the constant C is indeed undefined, and there might be an error in the solution process. Double-check your work and make sure that the steps were followed accurately.