Solving a Re-arranging Problem: Making r the Subject

[lostidentity]

I'm trying integrate the following equation and make r the subject
$$\frac{dr}{dt} = \Phi - \Psi \frac{2}{r}\frac{dr}{dt}$$

I first collect the derivative terms together and integrate the equation with respect to r and t to obtain

$$r + 2\Psi\ln{r} = \Phi{t} + r_0$$

where r0 is the constant of integration. My question is how would I make r the subject of the above equation?

Many thanks.

 

[tiny-tim]

hi lostidentity!

My question is how would I make r the subject of the above equation?

not possible!

(unless you define the LHS to be f(r), in which case it's r = f^-1(RHS) )

 

[lostidentity]

I'm wondering if there is another way to solve the ODE I gave in the previous post, i.e.

$$\left(1+2\frac{\Psi}{r}\right)\frac{dr}{dt} = \Phi$$

 

[Bill Simpson]

I'm wondering if there is another way to solve the ODE I gave in the previous post, i.e.

$$\left(1+2\frac{\Psi}{r}\right)\frac{dr}{dt} = \Phi$$

Check this very very carefully

$$r = 2 \Psi lambert\left(\frac{e^{\frac{t\Phi+c}{2 \Psi}}}{2 \Psi}\right)$$

where c is some constant and where lambert gives the principle solution for w in z=w e^w.

When I substitute this back into the original ODE it seems to check, but do not trust this until you have triple checked it.

 

[CellCoree]

To make r the subject of the equation, we can first isolate the term containing r on one side of the equation. In this case, we can subtract 2\Psi\ln{r} from both sides, giving us:

r = \Phi{t} + r_0 - 2\Psi\ln{r}

Next, we can use the property of logarithms to rewrite the equation as:

r = \Phi{t} + r_0 - \ln{r^{2\Psi}}

Finally, we can raise both sides to the power of e to eliminate the logarithm and obtain r as the subject:

e^r = e^{\Phi{t} + r_0 - \ln{r^{2\Psi}}}

r = e^{\Phi{t} + r_0} \cdot e^{-\ln{r^{2\Psi}}}

r = e^{\Phi{t} + r_0} \cdot \frac{1}{r^{2\Psi}}

Therefore, r is the subject of the equation and can be expressed in terms of the other variables. It is important to note that there may be multiple ways to manipulate an equation to make a specific variable the subject, and the method used may depend on the specific equation and its properties.