Answer by John Zangwill for Second order linear homogeneous ODE with constant...
This is an old question, but I don't believe that it was answered satisfactorily. What was wanted was an elementary direct proof showing that the usual solutions are the only solutions.Let...
View ArticleSecond order linear homogeneous ODE with constant coefficients
In homework I was asked to find all solutions to the following ODE:$$x''+ax'+bx = 0$$After reading, I know the following. (a)If $t^2+at+b = (t-\lambda_1)(t-\lambda_2)$ with $\lambda_1\ne\lambda_2$,...
View ArticleAnswer by dato datuashvili for Second order linear homogeneous ODE with...
you need to consider1.when determinant of characteristic equation (D) is positive ,in this case you have two real distinct root and colution is $c_1*e^{k_1*x}+c_2*e^{k_2*x}$where $k_1,k_2$ are roots of...
View ArticleAnswer by André Nicolas for Second order linear homogeneous ODE with constant...
One could quote a more general uniqueness theorem, and even prove it. But we will stick to the particular type of equation. And, in order to use only first-year calculus material, we will assume that...
View ArticleAnswer by mez for Second order linear homogeneous ODE with constant coefficients
Inspired by user1709828. Transform into first order ODE.$$y' = F(y) = (y(2), -a\cdot y(2)-b\cdot y(1))$$Claim $F$ is Lipschitz. Proof:$|F(x)-F(y)| = \sqrt{|x_2-y_2|^2+(a(x_2-y_2)+b(x_1-y_1))^2} =...
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