Find the directional derivative of f at the given point in the direction indicated by the angle theta. f(x,y)=ye^{-x}, (0,4), \theta=\frac{2\pi}{3}

Solve differential equation $y^{\prime }+y\cos x=\sin x\cos x$

Need your help!
1. Identify the substitution method used to solve the given differential equations. Support your answer with the appropriate calculations.
2. Solve for the solution to the given differential equation that satisfies the given condition
${\displaystyle y^{\prime }=2{(3x+y)}^2-1;y\left(0\right)=1}$

How can I find at least one first order differential equation when the solution for it is ${\displaystyle y=x^4}$?

Solve differential equation $dy/dx=15x^2{e}^{-y}$

Solve differential equation $t{y}^{\prime }+(t+1)y=t,y(\ln 2)=1$, t>0

Find the general solution of the first-order linear differential equation $(dy/dx)+(1/x)y=6x+2$, for x > 0

Classify the equation as separable, linear, both, or neither.
${\displaystyle y\frac{dy}{dx}=2x+y}$