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Let’s check out your problem:

(dx)/(dt)+(x)/(t^(2))=(1)/(t^(2))

dxdt+xt2=1t2 \frac{d x}{d t}+\frac{x}{t^{2}}=\frac{1}{t^{2}}

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Full solution

Q. dxdt+xt2=1t2 \frac{d x}{d t}+\frac{x}{t^{2}}=\frac{1}{t^{2}}
  1. Separate variables: Separate the variables by moving all terms involving xx to one side and all terms involving tt to the other side.
  2. Combine terms: Combine the terms on the right side to have a common denominator.
  3. Integrate with respect to tt: Integrate both sides with respect to tt.
  4. Integrate left side to xx: The left side integrates to xx. On the right side, we need to integrate term by term.
  5. Integrate first term: Integrate the first term on the right side.
  6. Use integrating factor: The second integral is not straightforward because it involves xx, which is a function of tt. This is a first-order linear differential equation, and we should use an integrating factor.

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