# Symbolic solution of first order differential equation

Is it possible to get simbolical solution of this differential equation: $$\dfrac{\mathrm{d}}{\mathrm{d}z}(x_0x_1) = -\dfrac{8}{r_i-z(r_i-1)}\dfrac{\mathrm{d} x_0}{\mathrm{d}z}$$ where if $$z=1: x_0=1, x_1=0$$. I tried to solve it like this, but it does not work:

sol = DSolve[
x1'[z] == (-8 * x0'[z]/(ri - z (ri - 1)) - x0'[z]*x1[z])/x0[z], x1,
z]


I tried to integrate it by hand, but I am making some mistakes because I can see that the numerical and my solution are different (probably something between $$\mathrm{d}z$$ and $$r_i-z(r_i-1)$$).

• Condition x1==0 and use inside DSolve as function x1[z] are contradictory! Apr 22, 2022 at 15:47
• Why is contradictory? Apr 22, 2022 at 15:49
• You use the same symbol as parameter and function! Apr 22, 2022 at 15:52
• Do you mean z is the same symbol as parameter and function? Or some other symbol? Apr 22, 2022 at 15:56
• Your initial conditions should be written as x0[1] == 1, x1[1] == 0; however, you have two functions with only one equation. Your system is underdetermined. Apr 22, 2022 at 15:56

Assumig you want solve the TeXform of the equation:

Integrate both sides

x0[z]x1[z]-x0[1]x1[1]== -Integrate[-8 /(ri - zz (ri - 1)) x0'[zz],{zz,1,z}]


Knowing x1[1]==0 it follows

x1[z]== -Integrate[-8 /(ri - zz (ri - 1)) x0'[zz],{zz,1,z}]/x0[z]

• Yes, but Mathematica does not give solution, in solution it just write Integrate, or integral symbol. Apr 22, 2022 at 16:03
• That's true, Mathematica provides a form for arbitrary function x0[z] Apr 22, 2022 at 16:05
• Well, I have x0[z], it is x0[z] = (1 + 64 beta (1 - 1/(ri - z (ri - 1))^3)/(3 (ri - 1)))^0.5. Now your x0[z]x1[z]==c-Integrate[-8 /(ri - z (ri - 1)) x0'[z],z] gives me soluton with Hypergepmetric function, so it probably means it is not so easy to integrate this by hand, as I thought? Apr 22, 2022 at 16:08
• What do you know about parameter ri and beta? Are there singularities of the integrand? Apr 22, 2022 at 16:24
• ri and beta are not dependent on z, they are constants. Apr 22, 2022 at 17:03