What is a solution to the differential equation $\frac{dy}{dx}=2{e}^{x-y}$ with the initial condition $y\left(1\right)=\mathrm{ln}(2e+1)$?

cubanwongux
2022-09-13
Answered

What is a solution to the differential equation $\frac{dy}{dx}=2{e}^{x-y}$ with the initial condition $y\left(1\right)=\mathrm{ln}(2e+1)$?

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I have a first order PDE:

$x{u}_{x}+(x+y){u}_{y}=1$

With the initial condition:

I have calculated result in Mathematica: $u(x,y)={\displaystyle \frac{y}{x}}$, but I am trying to solve the equation myself, but I had no luck so far. I tried with method of characteristics, but I could not get the correct results. I would appreciate any help or maybe even whole procedure.

$x{u}_{x}+(x+y){u}_{y}=1$

With the initial condition:

I have calculated result in Mathematica: $u(x,y)={\displaystyle \frac{y}{x}}$, but I am trying to solve the equation myself, but I had no luck so far. I tried with method of characteristics, but I could not get the correct results. I would appreciate any help or maybe even whole procedure.

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I have a first order differential equation question with initial value.

$\frac{dv}{dt}+\frac{{v}^{2}}{5}=9.8\text{with}v(0)=0$

I tried to solve it but just didn't know how to remove the absolute sign in my solution.

Please let me know if whole question is needed to remove the absolute sign inside the solution.

Edit 1: Part where I got stuck:

After integration and since C = 0:

$\frac{5ln(lv+7l)}{14}-\frac{5ln(lv-7l)}{14}=t$

Edit 2: I think my solution is wrong

$\frac{dv}{dt}+\frac{{v}^{2}}{5}=9.8\text{with}v(0)=0$

I tried to solve it but just didn't know how to remove the absolute sign in my solution.

Please let me know if whole question is needed to remove the absolute sign inside the solution.

Edit 1: Part where I got stuck:

After integration and since C = 0:

$\frac{5ln(lv+7l)}{14}-\frac{5ln(lv-7l)}{14}=t$

Edit 2: I think my solution is wrong

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