Question

Solve the differential equation. dy dy dx = e⁵ˣ -⁵ʸ

a.y= In (e⁵ˣ + C)
b. y = 5ln (e⁵ˣ + C)
c.y = 5e⁵ˣ + C
d.y= 1/5 in (e⁵ˣ + C)

Answer 1

If the differential equation is actually meant to be
`dy/dx = e^(5x)`, then after integrating both sides, applying the natural logarithm, and simplifying, the correct solution is option d
`: y = 1/5 ln(e^(5x) + C).`

Step-by-step explanation:

To solve the differential equation
`dy/dx = e^(5x) - 5^y`, we need to separate variables and integrate. However, if this is an error and the actual equation is
`dy/dx = e^(5x)` which is separable, or potentially
`dy/dx = e^(5x-5y)`which could be solved via separation of variables, we can find the solution presented in the student's options.

If the equation is
`dy/dx = e^(5x)`, we would integrate both sides with respect to x to obtain:

• 
  `∫ dy = ∫ e^(5x) dx`
• 
  `y = (1)/(5)e^(5x) + C`

Applying the natural logarithm, we get:

• 
  `y = 1/5 ln(e^(5x) + C)`

Hence, option d,
`y = 1/5 ln(e^(5x) + C)` would be correct if the differential equation is
`dy/dx = e^(5x).`