Question

Transform the given differential equation or system into an equivalent system of first-order differential equations. x''+2x'+26x=82\cos 4t

Exponential models
Transform the given differential equation or system into an equivalent system of first-order differential equations.
$$\displaystyle{x}{''}+{2}{x}'+{26}{x}={82}{\cos{{4}}}{t}$$

2021-07-05

Transform the given differential equation or system into an equivalent system of first-order differential equations.
$$\displaystyle{x}{''}+{2}{x}'+{26}{x}={82}{\cos{{4}}}{t}$$ The second-order equation $$\displaystyle{x}{''}+{2}{x}'+{26}{x}={82}{\cos{{4}}}{t}$$
is equivalent to system
$$f(t,x,x')=82\cos4t-26x-2x'$$
Hense the substitutions $$\displaystyle{x}_{{1}}={x},{x}_{{2}}={x}'={x}'_{{1}}$$
yield the system
$$\displaystyle{x}'_{{1}}={x}_{{2}}$$
$$\displaystyle{x}'_{{2}}={82}{\cos{{4}}}{t}-{26}{x}_{{1}}-{2}{x}_{{2}}$$ is a system of first-order equation.