How to solve a complicated ODEThe equation is f′(x)=γf(x)+f2(x)log⁡(f(x)1+f(x))with the initial condition f(0), where x≥ and f(0)≥0.The solution isf(x)=1−1+exp⁡2γx+log⁡(f(0)+1f(0))2I think the ODE can be solved by separating the variables∫0f(x)log⁡(f(z)1+f(z))f(z)+f2(z)df=∫0xγdzThe right-hand size is easy. I do not know how to solve the integration of the left-hand side.

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How to solve a complicated ODEThe equation is f′(x)=γf(x)+f2(x)log⁡(f(x)1+f(x))with the initial condition f(0), where x≥ and f(0)≥0.The solution isf(x)=1−1+exp⁡2γx+log⁡(f(0)+1f(0))2I think the ODE can be solved by separating the variables∫0f(x)log⁡(f(z)1+f(z))f(z)+f2(z)df=∫0xγdzThe right-hand size is easy. I do not know how to solve the integration of the left-hand side.

Roy Brady · Accepted Answer

Step 1Hint: Your method so far is correct. What you are asking for is the integral∫log⁡(y1+y)y+y2∼dy.Step 2The integral at first looks deceptively hard to solve but it is actually very easy if you notice it is of the form∫g(y)g′(y)∼dyfor g(y)=log⁡(y1+y).

Luciana Cline · Accepted Answer

As regards the integral on the left, note that∫1f(z)+f2(z)df=∫(1f(z)−11+f(z))df=log⁡(f(z)1+f(z))+c.

How to solve a complicated ODE The equation is

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