91Ó°ÊÓ

In Problems 15 and 16, interpret each statement as a differential equation. On the graph of \(y=\phi(x)\), the slope of the tangent line at a point \(P(x, y)\) is the square of the distance from \(P(x, y)\) to the origin.

Determine by inspection at least two solutions of the given first-order IVP. $$ y^{\prime}=3 y^{2 / 3}, \quad y(0)=0 $$

Verify that the indicated function \(y=\phi(x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example 5, by considering \(\phi\) simply as a function, give its domain. Then by considering \(\phi\) as a solution of the differential equation, give at least one interval \(I\) of definition. $$ (y-x) y^{\prime}=y-x+8 ; \quad y=x+4 \sqrt{x}+2 $$

A series circuit contains a resistor and a capacitor as shown. Determine a differential equation for the charge \(q(t)\) on the capacitor if the resistance is \(R\), the capacitance is \(C\), and the impressed voltage is \(E(t)\).

$$ x y^{\prime}=2 y, \quad y(0)=0 $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime}=25+y^{2} ; \quad y=5 \tan 5 x $$

Determine by inspection at least two solutions of the given first-order IVP. $$ x y^{\prime}=2 y, \quad y(0)=0 $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime}=2 x y^{2} ; \quad y=1 /\left(4-x^{2}\right) $$

Determine a region of the \(x y\)-plane for which the given differential equation would have a unique solution whose graph passes through a point \(\left(x_{0}, y_{0}\right)\) in the region. $$ \frac{d y}{d x}=y^{2 / 3} $$

(a) Give the domain of the function \(y=x^{2 / 3}\). (b) Give the largest interval \(I\) of definition over which \(y=x^{2 / 3}\) is a solution of the differential equation \(3 x y^{\prime}-2 y=0\).