Mathematics - Differential Equation Question with Solution | TestHub

A)

B) $\int \mathrm{f}(\mathrm{t})=\mathrm{dt}=10\int \mathrm{f}(\mathrm{t})=\mathrm{dt}=10\left(\frac{1}{2}\times 1\times \frac{1}{2}\right)=\frac{5}{2}=2.5$

C) $\frac{dx}{dt}=-f(t)x$ $\frac{dx}{x}=-f(t)dt$

$\mathrm{x}(\mathrm{t})={\lambda }_1{\mathrm{e}}^{-{\mathrm{t}}^2/2}$ if $0\le \mathrm{t}\le \frac{1}{2}$

$={\lambda }_2{\mathrm{e}}^{\frac{{\mathrm{t}}^2}{2}-\mathrm{t}}$ if $\frac{1}{2}\le \mathrm{t}\le 1$

$={\lambda }_3{\mathrm{e}}^{\mathrm{t}-{\mathrm{t}}^2/2}$ if $1\le \mathrm{t}\le \frac{3}{2}$

$={\lambda }_4{\mathrm{e}}^{\frac{{\mathrm{t}}^2}{2}-2\mathrm{t}}$ if $\frac{3}{2}\le \mathrm{t}\le 2$.

Which is clearly non periodic