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Chapter 12: Q31E (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.
\(y' + 2y = 0\)
The value of \(m\) is \( - 2\).
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In Problems \(13\) and \(14\) determine by inspection at least one solution of the given differential equation.
\(y' = y(y - 3)\)
In Problems \(1 - 8\) state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with \((6)\).
\(x\frac{{{d^3}y}}{{d{x^3}}} - {\left( {\frac{{dy}}{{dx}}} \right)^4} + y = 0\)
In Problems \(11 - 14\) verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.
\(y'' + y = tanx; y = - (cosx) ln(secx + tanx)\).
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 21鈥�24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.
\({x^3}\frac{{{d^3}y}}{{d{x^3}}} + 2{x^2}\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + y = 12{x^2};y = {c_1}{x^{ - 1}} + {c_2}x + {c_3}xlnx + 4{x^2}\)
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