Differential Equations 2024 Test with A Graded Solutions

Differential Equations 2024 Test with A Graded Solutions Seperable Differential Equation - Answer-(dy/dt)=g(t)h(y) Autonomous Differential Equation - Answer-(dy/dt)=h(y) Linear Differential Equation - Answer-(dy/dt)=a(t)y+b(t) Homogeneous Differential Equation - Answer-(dy/dt)=a(t)y Non-Homogeneous Differential Equation - Answer-(dy/dt)=a(t)y+b(t), where b(t) does not equal zero Linearity Principle - Answer-If y(t) is a solution to a homogeneous Differential Equation, then any constant multiple of that solution, ky(t), is also a solution. Extended Linearity Principle, Part 1 - Answer-If y_h(t) is any solution to the homogeneous equation and y_p(t) is any particular solution to the non-homogeneous equation, then y_h(t)+y_p(t) is a solution to the non-homogeneous equation. Extended Linearity Principle, Part 2 - Answer-If y_p and y_q are any two solutions to the non-homogeneous equation, then y_p(t)-y_q(t) is a solution to the associated homogeneous differential equation. Steady State Solution - Answer-A non-equilibrium solution that behaves like an equilibrium solution. Integrating Factors - Answer-y(t)=(1/u(t))int(u(t)b(t))dt , u(t)=e^(int(g(t))dt) Existence Theorem - Answer-Suppose f(t,y) is a continuous function in a rectangle of the form {t,y | atb, cyd} in the ty-plane. If (t_0, y_0) is a point in this rectangle, then there exists some E0 and a function y(t) defined for t_0-E t t_0+E that solves the associated IVP.