Deriving monotony via non-solvable differential equation

I have a differential equation which I think is pretty much unsolvable analytically and numerically, as I do not know $$y(c_0)$$. $$y(c_0)$$ is the value of $$y(x)$$ at $$c_0$$, however I do not know this value.

$$y'(x)=x y(x)^n+\frac{1}{x}(y(c_0)-y(x))^{-q}-\alpha y(x)$$

Can I deduce from this equation that $$y(x)$$ will be decreasing in $$\alpha$$? If yes, why? Can I even deduce whether $$y(x)$$ is decreasing or increasing in $$x$$?

• What does $y(c_0)$ represent, is it related to an initial condition? – WalterJ Aug 2 at 9:24
• Basically yes, but unfortunately I do not know the value, but $y'(x)$ depends on $y(c_0)$ – Paul Aug 2 at 9:37

If there are other specifications that allow you to show that $$y'$$ is always positive or negative for all $$x$$, then you can infer that $$y$$ Is monotone increasing or decreasing with $$x$$. But to determine how $$y'$$ changes with $$\alpha$$, you'd need to know the magnitude of the two other terms relative to $$\alpha y$$.

Just for the sake of experimentation, this MATLAB code integrates some version of your equation (note, I took $$y(c_0)=y_0$$). For this example $$y\to 0$$, but this is not generally true!

%% ODE example
clear all; clc; close all;
n=2;
q=-1;
a=-2;
t0 = 1;
t1 = 100;
tspan = [t0 t1];
y0 = -100;
[t,y] = ode23s(@(t,y) t*(y^n)+(1/t)*(y0-y)^(-q)-a*y, tspan, y0);
plot(t,y,'-k')
grid on

• Thank you Amy and walter – Paul Aug 2 at 14:40