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The solution of the differential equation dt/dx = t * (g'(x) - t²) / g(x) is:
- t = (g(x) + c) / x
- t = g(x)/x + c
- t = g(x) / (x + c)
- t = g(x) + x + c
Correct answer: t = g(x) / (x + c)
Solution
The given ODE dt/dx = t*g'(x)/g(x) - t³/g(x) is a Bernoulli equation of the form dt/dx - t*g'(x)/g(x) = -t³/g(x). Dividing by t³ and substituting v = t^(-2) linearises it. The resulting linear ODE dv/dx + 2*g'(x)/g(x)*v = 2/g(x) has integrating factor g(x)². Solving gives t = g(x)/(x+c).
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