find the general solution to the differential equation.
dy/dx=secxtanx-e^x
The equation is separable, meaning that we can put x and y on each side of the equal sign.
dy/dx=secxtanx-e^x
=>
dy = (sec(x)tan(x) - e^x)dx
Integrate both sides:
∫dy = ∫(sec(x)tan(x) - e^x)dx
=>
y = sec(x)-e^x + C
where C is an integration constant.
differentiate the above solution to confirm that the solution is correct.
To find the general solution to the given differential equation, we need to separate the variables and integrate both sides.
First, let's rewrite the given equation:
dy/dx = sec(x)tan(x) - e^x
Rearranging, we get:
dy = (sec(x)tan(x) - e^x)dx
Now, let's integrate both sides.
∫dy = ∫(sec(x)tan(x) - e^x)dx
The integral of dy is simply y, and the integral of e^x is e^x. However, the integral of sec(x)tan(x) is a bit more involved.
To integrate sec(x)tan(x), we can use a substitution. Let's let u = sec(x), then du = sec(x)tan(x)dx.
Now, the integral becomes ∫(sec(x)tan(x) - e^x)dx = ∫(du - e^x) = u - e^x.
Substituting back, we have:
y = u - e^x
= sec(x) - e^x
Therefore, the general solution to the given differential equation is y = sec(x) - e^x.