Find the equation of the curve

[karush]

Find the equation of the curve that passes through the point\(\displaystyle (1,2)\)
and has a slope of $\displaystyle\left( 3+\frac{1}{x}\right)$
at any point \(\displaystyle (x,y)\) on the curve.

$(A) 2xe^{3x-3}$
$(B) 2xe^{3x+3}$
$(C) 2xe^3$
$(D) 2e^{3x-3}$

\(\displaystyle A(1)=2\) and \(\displaystyle D(1)=2 \)
so \(\displaystyle B\) and \(\displaystyle C\) are out.

the answer is\(\displaystyle (A)\) but I took \(\displaystyle A'\) and \(\displaystyle D' \)but couldn't equate that to
$\displaystyle\left( 3+\frac{1}{x}\right)$

 

[SuperSonic4]

Find the equation of the curve that passes through the point\(\displaystyle (1,2)\)
and has a slope of $\displaystyle\left( 3+\frac{1}{x}\right)$
at any point \(\displaystyle (x,y)\) on the curve.

$(A) 2xe^{3x-3}$
$(B) 2xe^{3x+3}$
$(C) 2xe^3$
$(D) 2e^{3x-3}$

\(\displaystyle A(1)=2\) and \(\displaystyle D(1)=2 \)
so \(\displaystyle B\) and \(\displaystyle C\) are out.

the answer is\(\displaystyle (A)\) but I took \(\displaystyle A'\) and \(\displaystyle D' \)but couldn't equate that to
$\displaystyle\left( 3+\frac{1}{x}\right)$

edit: What Mark said below, go look at his post if you don't want confused ramblings (Giggle)

I get \(\displaystyle y = 3x + \ln|x| -1\) by integrating the slope and using the given point. Not sure where they're getting the exponential values from.

\(\displaystyle \frac{d}{dx} 2xe^{3x-3} = 2e^{3x-3} + 6x(x-1)e^{3x-3} = 2e^{3x-3}(1+3x(x-1))\) which does give 2 when I plug in 1

 

[MarkFL]

This is an IVP, where:

\(\displaystyle \frac{dy}{dx}=3+\frac{1}{x}\) and \(\displaystyle y(1)=2\)

Integrating the ODE, we obtain:

\(\displaystyle y(x)=3x+\ln|x|+C\)

Using the initial value, we may find $C$:

\(\displaystyle y(1)=3+C=2\,\therefore\,C=-1\)

And so the solution is:

\(\displaystyle y(x)=3x+\ln|x|-1\)

It appears to me that answer A) is the solution to:

\(\displaystyle y'=y\left(3+\frac{1}{x} \right)\)

 

[karush]

https://www.physicsforums.com/attachments/2279

not sure why they (AP calc exam) chose those curves?

thanks much for help..

 

[MarkFL]

Are you sure you copied the problem correctly and in its entirety here?