`(dr)/(dt) = (1+e^t)^2/e^(3t)` Solve the differential equation.

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`(dr)/dt=(1+e^t)^2/e^(3t)`

`r=int(1+e^t)^2/e^(3t)dt`

`r=int(1+2(1)e^t+(e^t)^2)/e^(3t)dt`

`r=int(1+2e^t+e^(2t))/e^(3t)dt`

`r=int(1/e^(3t)+(2e^t)/e^(3t)+e^(2t)/e^(3t))dt`

`r=int(e^(-3t)+2e^(t-3t)+e^(2t-3t))dt`

`r=int(e^(-3t)+2e^(-2t)+e^(-t))dt`

Apply the sum rule and take the constants out,

`r=inte^(-3t)dt+2inte^(-2t)dt+inte^(-t)dt`

Now use the common integral: `inte^x=e^x`

`r=e^(-3t)/(-3)+2e^(-2t)/(-2)+e^(-t)/(-1)`

simplify and add a constant C to the solution,

`r=-1/3e^(-3t)-e^(-2t)-e^(-t)+C`

`r=-(1/3e^(-3t)+e^(-2t)+e^(-t))+C`

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`(dr)/dt=(1+e^t)^2/e^(3t)`

`r=int(1+e^t)^2/e^(3t)dt`

`r=int(1+2(1)e^t+(e^t)^2)/e^(3t)dt`

`r=int(1+2e^t+e^(2t))/e^(3t)dt`

`r=int(1/e^(3t)+(2e^t)/e^(3t)+e^(2t)/e^(3t))dt`

`r=int(e^(-3t)+2e^(t-3t)+e^(2t-3t))dt`

`r=int(e^(-3t)+2e^(-2t)+e^(-t))dt`

Apply the sum rule and take the constants out,

`r=inte^(-3t)dt+2inte^(-2t)dt+inte^(-t)dt`

Now use the common integral: `inte^x=e^x`

`r=e^(-3t)/(-3)+2e^(-2t)/(-2)+e^(-t)/(-1)`

simplify and add a constant C to the solution,

`r=-1/3e^(-3t)-e^(-2t)-e^(-t)+C`

`r=-(1/3e^(-3t)+e^(-2t)+e^(-t))+C`

 

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