# Solving differential equation in The Economics of Superstar (by Rosen)

I am currently reading "The Economics of Superstar" written by Rosen (1981). I don't understand one differential equation he used in the paper. The equation is as follows: $$\frac{dp}{dz} = (p+s)/z.$$

He says if we integrate this, we get $p(z) = vz-s$ for $v = (p+s)/z$. I think that it is a simple differential equation, but I am having trouble solving this. Could you give some help?

• Hi: There may be an easier way (my knowledge is that I took a course in dfeqs 30 years ago ) but, if you re-write it as $dp/dz - p/z = s/z.$, then, it is of the form $p^{\prime} - P(z) p = Q(z)$ and you can use the integrating factor method shown on page 11 of this link. math.hawaii.edu/~jamal/tuc01alt_desupps.pdf Aug 8 '18 at 4:49

The differential equation is of the form

$$y' + f(x)y = q(x)$$

The correct answer in our case is

$$p = -s$$

so that you know what you are targeting. Namely, it does not depend on $z$. You can verify that it satisfies the differential equation.

Then the author just plays around like

$$p= - s \implies p -p = s-s \implies p - \frac p z z = \frac s z z - s$$

$$p = \frac p z z + \frac s z z - s = \frac {p+s}{z}z - s$$

$$\implies p = vz -s$$

...and I suppose this playing around with identities is helpful to what he does in the paper.

• Hi Alecos: Two questions at your convenience of course. 1) If you wouldn't mind, could you define your $y^{\prime}$, $p(x)$ and $q(x)$ terms so that I can understand how mine was wrong. 2) Where did you get $p = -s$ from ? thanks. Aug 9 '18 at 14:17
• @markleeds There's nothing wrong with what you wrote. I wrote exactly the same thing that you did. Solving this differebtial equation, we get as solution $p=-s$. For the solution , see for example, mathworld.wolfram.com/… Aug 9 '18 at 14:19
• Thanks. I'll check it out. Solving it looked hazardous so I didn't try !!!!! Aug 9 '18 at 14:22