In we saw that cos x and ex were solutions of the nonlinear equation ( y) 2 y2 0. Verify
Chapter 3, Problem 20(choose chapter or problem)
Discussion Problems
In Problem 1 we saw that cos x and \(e^{x}\) were solutions of the nonlinear equation \(\left(y^{\prime \prime}\right)^{2}-y^{2}=0\). Verify that sin x and \(e^{-x}\) are also solutions. Without attempting to solve the differential equation, discuss how these explicit solutions can be
found by using knowledge about linear equations. Without attempting to verify, discuss why the linear combinations \(y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} \cos x+c_{4} \sin x\) and \(y=c_{2} e^{-x}+c_{4} \sin x\) are not, in general, solutions, but the two special linear combinations \(y=c_{1} e^{x}+c_{2} e^{-x}\) and \(y=c_{3} \cos x+c_{4} \sin x\) must satisfy the differential equation.
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