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

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Solve the following first order differential equations. Use any method:
${\displaystyle 2x^{2}dx-3x{y}^{2}dy=0;y\left(1\right)=0}$

Solve differential equation $(x+2)y\prime +4y=(3x+6{)}^{-}2\ln x$

Suppose we have
${\displaystyle \frac{dy}{dx}+f\left(x\right)y=r\left(x\right)}$
and it has two solutions ${\displaystyle y_1\left(x\right)}$ and ${\displaystyle y_2\left(x\right)}$ then how to prove that solution of differential equation
${\displaystyle \frac{dy}{dx}+f\left(x\right)y=2r\left(x\right)}$
Will be ${\displaystyle y_1\left(x\right)+y_2\left(x\right)}$? I think given differential equations is linear first order equation so its solution will be
${\displaystyle y.e^{\int f\left(x\right)dx}=\int r.e^{\int f\left(x\right)dx}dx}$
now do I establish two solution as ${\displaystyle y_1}$ and ${\displaystyle y_2}$ out of this equation?

Is there any possibility to convert first order differential equation to second order differential equation?
I have a system of first order differential equations as below and i need the right hand side of the second order form of this first order equation. (if possible)
$\frac{du(t)}{dt}=[t{u}_2(t);4u_1(t{)}^{3/2}]$
Any help would be appreciated.

Solve and find the soltion to the first order differential equation:
${\displaystyle x\frac{dy}{dx}+y=e^x,x>0}$

Solve differential equation $dy/dx=x/(y{e}^{x+y^2})$