Question

Give the general solution of

y'=2e²ˣ⁻ʸ.
a.y=e²ˣ+C
b.e⁻ʸ=2e²ˣ+C
c.eʸ=Ce²ˣ
d.y=ln(e²ˣ+C)
e.y=2ln(1/2e²ˣ+C)
f.None of the above.

Answer 1

The general solution of the differential equation y'=2e^{2x-y} is d. y = ln(e^{2x} + C), which requires separating variables and integrating both sides.

Step-by-step explanation:

The student is asked to find the general solution of the differential equation y'=2e2x-y. To solve this, we need to separate the variables and integrate. The differential equation can be rewritten as dy/ey = 2e2xdx. Integrating both sides gives ∫dy/ey = ∫2e2xdx, which upon integration yields -e-y = e2x+C. Multiplying both sides by -1 gives e-y = -e2x-C. Taking the natural logarithm of both sides to solve for y, we get y = -ln(e2x + C). To match the options available, we can absorb the negative sign into the constant, so the general solution is d. y = ln(e2x + C).