2계 선형 제차 미분방정식

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$25\frac{{d}^{2}y}{d{x}^{2}}-80\frac{dy}{dx}+64y=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$\frac{{d}^{2}y}{d{x}^{2}}-9\frac{dy}{dx}=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$4\frac{{d}^{2}y}{d{x}^{2}}+8y=0$

(Principle of superposition) Prove that if ${y}_{1}(x)$ and ${y}_{2}(x)$ are solutions to a linear homogeneous differential equation, ${y}^{''}+p(x){y}^{'}+q(x)y=0,$ then the function $y(x)={c}_{1}{y}_{1}(x)+{c}_{2}{y}_{2}(x),$ where ${c}_{1}$ and ${c}_{2}$ are constants, is also a solution.

Prove that if a, b, and c are positive constants, then all solutions to the second-order linear differential equation $a{y}^{''}+b{y}^{'}+cy=0$ approach zero as $x\to \infty .$ (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)

Assuming the system starts from rest, show that the particular solution can be written as $y=\frac{2}{a({\omega }_{0}{}^{2}-{\omega }^{2})}\sin \left(\frac{{\omega }_{0}-\omega t}{2}\right)\sin \left(\frac{{\omega }_{0}+\omega t}{2}\right).$