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Chapter 12: Q37E (page 518)
In Problems \(37 - 40\) use the concept that \(y = c, - \infty < x < \infty \), is a constant function if and only if \(y' = 0\) to determine whether the given differential equation possesses constant solutions.
\(3xy' + 5y = 0\)
There exists a constant solution (i.e.) \(y = 2\).
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In Problems \(35\) and \(36\) find values of m so that the function \(y = {m^x}\) is a solution of the given differential equation.
\(xy'' + 2y' = 0\)
(a) Give the domain of the function \(y = {x^{{\textstyle{2 \over 3}}}}\).
(b) Give the largest interval I of definition over which \(y = {x^{{\textstyle{2 \over 3}}}}\) is a solution of the differential equation \(3xy' - 2y = 0\).
In Problems \(35\) and \(36\) find values of m so that the function \(y = {m^x}\) is a solution of the given differential equation.
\({x^2}y'' - 7xy' + 15y = 0\)
In Problems 25鈥�28 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.
\({x^2}\frac{{dy}}{{dx}} + xy = 10sinx;y = \frac{5}{x} + \frac{{10}}{x}\int_1^x {\frac{{sint}}{t}} dt\)
In Problems 7鈥�12 match each of the given differential equations with one or more of these solutions:
(a) \(y = 0\), (b) \(y = 2\), (c) \(y = 2x\), (d) \(y = 2{x^2}\)
\(y' = 2\)
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