Start by separating variables:
\(\frac{dx}{dt} = \frac{1}{x} - \frac{x}{4}\)
Rearrange to:
\(x \frac{dx}{dt} = 1 - \frac{x^2}{4}\)
Separate variables:
\(\frac{x}{1 - \frac{x^2}{4}} dx = dt\)
Integrate both sides:
\(\int \frac{x}{1 - \frac{x^2}{4}} dx = \int dt\)
Let \(u = 1 - \frac{x^2}{4}\), then \(du = -\frac{x}{2} dx\).
Substitute and integrate:
\(-2 \int \frac{1}{u} du = t + C\)
\(-2 \ln|u| = t + C\)
Substitute back for \(u\):
\(-2 \ln|1 - \frac{x^2}{4}| = t + C\)
Use initial condition \(x = 1\) when \(t = 0\):
\(-2 \ln|1 - \frac{1}{4}| = 0 + C\)
\(-2 \ln\left(\frac{3}{4}\right) = C\)
\(C = 2 \ln 3\)
Substitute \(C\) back:
\(-2 \ln|1 - \frac{x^2}{4}| = t + 2 \ln 3\)
Rearrange to solve for \(x^2\):
\(\ln|1 - \frac{x^2}{4}| = -\frac{1}{2}t - \ln 3\)
\(|1 - \frac{x^2}{4}| = 3\exp\left(-\frac{1}{2}t\right)\)
\(1 - \frac{x^2}{4} = 3\exp\left(-\frac{1}{2}t\right)\)
\(x^2 = 4 - 3\exp\left(-\frac{1}{2}t\right)\)
