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 ) $

  • $\begingroup$ It is a Riccati equation and it can be reduced to a second order linear equation. $\endgroup$ – Gribouillis Nov 12 '17 at 14:34

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.