Is $y=-2x+2.5$ a singular solution to the differential equation $y'=(4x+2y-1)^{1/2}?$

Question

I have a hunch that $y=-2x+2.5$ is a singular solution to the differential equation $$y'=(4x+2y-1)^\frac{1}{2}$$But I don't know if I'm right. Furthermore, I don't know how to check if I'm right. How do I proceed?

Answer 1

• First of all set $t=4x+2y-1$ and then try to write the OE with respect to this new substitution to find the following ODE: $$\frac{dt}{dx}=2\sqrt{2}+4$$.

• Solve the later OE.

• Assuming the function $y=-2x+2.5$ is the solution of the original OE, check if this function can be achieved from the one-parameter family of solutions of the original OE.

If $y$ cannot be achieved by putting some $C_1$ in the family of solutions, then it is a singular solution.