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I am a beginner in Mechanical Engineering research field. Recently I read a paper that contains one non-linear ordinary differential equation. The original author's have not explained anything about how the non-linear ordinary differential equation has been solved. I request you people to kindly help me by providing solution steps or any link that can help me to find solution of the same. The equation is:

$\frac{dQ}{dT} + Q^2 - \frac{1}{({\beta}T+1)^2}=0$

and solution of the above equation as suggested by the original authors is: $Q(t) = \frac{1}{2({\beta}T+1)} \left ( \beta + tanh\left ( \frac{1}{2}.\frac{ln({\beta}T+1)\sqrt{{\beta}^2 + 4}-2.arctanh(-2+{\beta})/\sqrt{{\beta}^2 + 4}.\beta}{\beta} \right ) \sqrt{{\beta}^2 + 4} \right ) $

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  • $\begingroup$ It is a Riccati equation and it can be reduced to a second order linear equation. $\endgroup$ Nov 12, 2017 at 14:34

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You have $$\frac{dQ}{dT}+Q^{2}=\frac{1}{(\beta{T}+1)^{2}}$$ Let $$Q(T)=\frac{A}{(\beta{T}+1)}$$ Then $$\frac{dQ}{dT}+Q^{2}=\frac{A^{2}-\beta{A}}{(\beta{T}+1)^{2}}$$ Thus, for the solution, we want $A^{2}-\beta{A}-1=0$, this gives you two particular solutions $Q_{p, 1/2}$. Two more general solution containing a single arbitrary constant each can be obtained by then letting $$Q_{1/2}=Q_{p, 1/2}+\frac{1}{y_{1/2}}$$ This will give two linear equations for $y_{1/2}$, and they are easy to solve.

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