Chapter 1: IntroductionContent check — Differential Equations (Lebl), Chapter 1: Introduction

Show that $x=e^{-2t}$ is a solution to $x^{\prime \prime }+4x^{\prime }+4x=0$⁠.

Is $y=x^2$ a solution to $x^2{y}^{\prime \prime }-2y=0$⁠? Justify.

Let $x{y}^{\prime \prime }-y^{\prime }=0$⁠. Try a solution of the form $y=x^r$⁠. Is this a solution for some $r$⁠? If so, find all such $r$⁠.

Verify that $x=C_1{e}^t+C_2$ is a solution to $x^{\prime \prime }-x^{\prime }=0$⁠. Find $C_1$ and $C_2$ so that $x$ satisfies $x(0)=10$ and $x^{\prime }(0)=100$⁠.

Solve $\frac{d\phi }{ds}=8\phi$ and $\phi (0)=-9$⁠.

Solve:

a) ${\displaystyle \frac{dx}{dt}}=-4x,x(0)=9$

b) ${\displaystyle \frac{d^{2}x}{d{t}^2}}=$ $-$ $4x,$ $x(0)=$ $1,$ $x^{\prime }(0)=$ $2$

c) ${\displaystyle \frac{dp}{dq}}=3p,p(0)=4$

d) ${\displaystyle \frac{d^{2}T}{d{x}^2}}=$ $4T,$ $T(0)=$ $0,$ $T^{\prime }(0)=$ $6$

Classify the following equations. Are they ODEs or PDEs? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant-coefficient? If it is an ODE, is it autonomous?

a) ${\displaystyle \frac{{\partial }^{2}v}{\partial x^2}+3\frac{{\partial }^{2}v}{\partial y^2}=\sin (x)}$

b) ${\displaystyle \frac{dx}{dt}+\cos (t)x=t^2+t+1}$

c) ${\displaystyle \frac{d^{7}F}{d{x}^7}=3F(x)}$

d) ${\displaystyle y^{\prime \prime }+8y^{\prime }=1}$

e) ${\displaystyle x^{\prime \prime }+ty{x}^{\prime }=0,y^{\prime \prime }+txy=0}$

f) ${\displaystyle \frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial s^2}+u^2}$

Write down the general zeroth-order linear ordinary differential equation. Write down the general solution.

For which $k$ is $\frac{dx}{dt}+x^k=t^{k+2}$ linear? Hint: There are two answers.