Can This Differential Equation Be Solved by Separation of Variables?

[Simfish]

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Okay, so I tried to separate the (1-W/b) term by letting it go into the denominator. But then you can't really solve the equation by the traditional dW/W method, since W + constant is raised to a power of n. i could multiply both sides by (1-W/b) to a power of n-1, but then when i integrate, i don't get the desired answer. can anyone help? thanks

 

[HallsofIvy]

You have
$$\frac{dW}{dz}= \frac{M}{(1-\frac{z}{a})(1-\frac{W}{b})^n}$$ W(0)= 0

That separates as
$$(1- \frac{W}{b})^n dW= \frac{M dz}{(1-\frac{z}{a})}$$
I'm not sure what you mean by "the traditional dW/W method" but if you mean just integrating to get ln(W), that works in only a tiny fraction of such separable differential equations. To integrate the left side, I would recommend letting u= 1- W/b so that du= -dW/b and the integral becomes
$$\int (1- \frac{W}{b})^n dW= -\frac{1}{b}\int u^n du$$