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Chapter 12: Q32E (page 517)
In Problems \(31 - 34\) find values of m so that the function \(y = m{e^{mx}}\) is a solution of the given differential equation.
\(5y' = 2y\)
The value of \(m\) is \(\frac{2}{5}\).
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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.
\(y' = {y^2} + 2y - 3\)
In Problems \(13\) and \(14\) determine by inspection at least one solution of the given differential equation.
\(y'' = y'\)
In Problems \(19\) and \(20\) verify that the indicated expression is an implicit solution of the given first-order differential equation. Find atleast one explicit solutionin each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi \).
\(2xydx + ({x^2} - y)dy = 0; - 2{x^2}y + {y^2} = 1\)
What is the slope of the tangent line to the graph of a solution of \(y' = 6\sqrt y + 5{x^3}\) that passes through \(( - 1,4)\)?
In Problems \(15 - 18\) verify that the indicated functionis an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.
\(y' = 25 + {y^2};y = 5tan5x\)
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