91Ó°ÊÓ

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ \left(\frac{e^{-2 \sqrt{x}}-y}{\sqrt{x}}\right) \frac{d x}{d y}=1, \quad y(1)=1 $$

Find a solution of \(x \frac{d y}{d x}=y^{2}-y\) that passes through the indicated points. (a) \((0,1)\) (b) \((0,0)\) (c) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (d) \(\left(2, \frac{1}{4}\right)\)

A critical point \(c\) of an autonomous first-order \(\mathrm{DE}\) is said to be isolated if there exists some open interval that contains \(c\) but no other critical point. Discuss: Can there exist an autonomous DE of the form given in (1) for which every critical point is nonisolated? Do not think profound thoughts.

A critical point \(c\) of an autonomous first-order DE is said to be isolated if there exists some open interval that contains \(c\) but no other critical point. Discuss: Can there exist an autonomous DE of the form given in (1) for which every critical point is nonisolated? Do not think profound thoughts.

Solve the given initial-value problem. Give the largest interval \(\boldsymbol{l}\) over which the solution is defined. $$ \left(1+t^{2}\right) \frac{d x}{d t}+x=\tan ^{-1} t, \quad x(0)=4 $$

A 200-volt electromotive force is applied to an \(R C\)-series circuit in which the resistance is 1000 ohms and the capacitance is \(5 \times 10^{-6}\) farad. Find the charge \(q(t)\) on the capacitor if \(i(0)=0.4\). Determine the charge and current at \(t=0.005 \mathrm{~s}\). Determine the charge as \(t \rightarrow \infty\).

Solve the given differential equation by finding, as in Example 4 , an appropriate integrating factor. $$ y(x+y+1) d x+(x+2 y) d y=0 $$

Find a solution of \(x \frac{a y}{d x}=y^{2}-y\) that passes through the indicated points. (a) \((0,1)\) (b) \((0,0)\) (c) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (d) \(\left(2, \begin{array}{l}1 \\ 4\end{array}\right)\)

$$ \begin{aligned} &\left(1+t^{2}\right) \frac{d x}{d t}+x=\tan ^{-1} t, \quad x(0)=4\\\ &\text { [Hint: In your solution let } \left.u=\tan ^{-1} t .\right] \end{aligned} $$

An electromotive force $$ E(t)= \begin{cases}120, & 0 \leq t \leq 20 \\ 0, & t>20\end{cases} $$ is applied to an \(L R\)-series circuit in which the inductance is 20 henries and the resistance is \(2 \mathrm{ohms}\). Find the current \(i(t)\) if \(i(0)=0\).