91Ó°ÊÓ

Show that \(x(t)=C e^{-t}+\frac{1}{2} e^{t}\) is a solution of the differential equation \(\dot{x}(t)+x(t)=e^{t}\) for all values of the constant \(C\).

Find the general solutions of the following differential equations. Also find the integral curves through the indicated points. (a) \(t \dot{x}=x(1-t),\left(t_{0}, x_{0}\right)=(1,1 / e)\) (b) \(\left(1+t^{3}\right) \dot{x}=t^{2} x,\left(t_{0}, x_{0}\right)=(0,2)\) (c) \(x \dot{x}=t,\left(t_{0}, x_{0}\right)=(\sqrt{2}, 1)\) (d) \(e^{2 t} \dot{x}-x^{2}-2 x=1,\left(t_{0}, x_{0}\right)=(0,0)\)