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I have Newton's second law equation: $F=ma$ where I consider mass $m$ as a constant. The net force $F$ is a sum of two forces $F_a$ and $F_b$ acting on it. $F_a$ is constant and $F_b$ varies according to $u(t)$.

So, the entire equation is, $F_a - F_b(u) = m \dfrac{du}{dt}$, which resembles the form $u' = a - bu$ where $a=F_a$ and $b=F_b$. How can make use of an educational guess (Ansatz form) and proceed for the above differential equation to solve for $u(t)$?

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    Expression for $F_b$ is needed to give an educated guess. – 2017-02-14

1 Answers 1

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The equation is separable.

$$m\frac{du}{F_a-F_b(u)}=dt,$$ then $$m\int_{u_0}^u\frac{du}{F_a-F_b(u)}=t.$$

After integration, you will need to invert the antiderivative.