So as I understand, the super position principle means that if we have several particular solutions to a DE we can add them together to get a general solution. Am I missing anything there? Like [math]y_1\:=\:x^2\:and\:y_2\:=\:e^x\:\:->\:y_g=c_1x^2+c_2e^x[/math] I don't know if there are any limitations here, but it seems that any solution can be added together to create a general solution respective to a DE. Now for the fundamental set this is taken a little bit further. If those solutions are... Read more ..read more

In the photo above why can't leave it as e^t(a-s)? Why do we need to take out a negative to convert it to e^-t(s-a ..read more

Dear expert, I would greatly appreciate it if someone could review the following question and answer. Please point out any errors in the answer, and let me know if there are any mistakes in my approach to solving the question. Thank you ..read more

Hello, I see many examples for solutions of the heat problem in cylindrical coordinates. I notice one thing in common. They all apply to a solid cylinder. In this case the solution after separating variables gives, separation equation u(r,t) = RT gives R(r) = A*J(0,L*r)+B*Y(0,L*r) and T(t) = C*e^(-alpha*L^2*t) In the typical examples it states B = 0 due to the radius r = 0 of the solid cylinder. therefore, R(r) = A*J(0,L*r) and I can the obtain my lambdas written here as L based... Read more ..read more

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I don't understand how he goes from C1*e+C2*e^2 to 1/e-e^2 ..read more

Okay so I rewrite it as [math]y'' + [(y')^2 - 1]y' + y = 0[/math] So if I understand right damping is energy lost from the oscillation. So if there is no damping there will be an infinite oscillation. With that thinking a negative damping will cause an increasing amplitude of the sinusoid. 1.) When y'=-1,1 we get infinite oscillation. 2.) When -1 < y' < 1 we get negative damping and therefor increasing amplitude. 3.) When -1 > y' > 1 we get positive damping and therefor decreasing... Read more ..read more

My first instinct was to use the Energy Integral Lemma given from my book [math]t = \pm \int \frac{1}{\sqrt{2 (F(x) + K)}} dy + c[/math] K being a constant Which I feel like is sufficient but the solution manual provides the following. Which I don't understand and didn't even know that the chain rule could be used on function notation like this. I guess the s is a stand in for -t. Crazy that you can use the chain rule like this. Is what I did sufficient or... Read more ..read more