Question

If dy/dt=-2y and if y=1 when t=0, what is the value of t for which y=1/2

Answer 1

$\bf \displaystyle \int -2t\cdot dt\implies -2\int t\cdot dt\implies -2\cdot \cfrac{t^2}{2}+C \\\\\\ -t^2+C=f(x)\qquad \begin{cases} f(x)=y=1\\ t=0 \end{cases}\implies -0^2+C=1\implies C=1 \\\\\\ -t^2+1=f(x)\\\\ -----------------------------\\\\ -t^2+1=f(x)\qquad f(x)=\cfrac{1}{2}\implies -t^2+1=\cfrac{1}{2} \\\\\\ 1-\cfrac{1}{2}=t^2\implies \pm \sqrt{\cfrac{1}{2}}=t\implies \cfrac{1}{\pm \sqrt{2}}=t \\\\\\ \pm \cfrac{\sqrt{2}}{2}=t\impliedby \textit{with a rationalized denominator}$