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
ANSWERED
asked 2021-07-04
Transform the given differential equation or system into an equivalent system of first-order differential equations.
\(\displaystyle{x}{''}+{2}{x}'+{26}{x}={82}{\cos{{4}}}{t}\)

Expert Answers (1)

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.

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