Question

# Transform the given differential equation or system into an equivalent system of first-order differential equations. x^{(4)}+3x''+x=e^{t}\sin 2t

Exponential models
Transform the given differential equation or system into an equivalent system of first-order differential equations.
$$\displaystyle{x}^{{{\left({4}\right)}}}+{3}{x}{''}+{x}={e}^{{{t}}}{\sin{{2}}}{t}$$

2021-05-27

Transform the given differential equation or system into an equivalent system of first-order differential equations.
$$\displaystyle{x}^{{{\left({4}\right)}}}+{3}{x}{''}+{x}={e}^{{{t}}}{\sin{{2}}}{t}$$ The fourth-order equation $$\displaystyle{x}^{{{\left({4}\right)}}}+{3}{x}{''}+{x}={e}^{{{t}}}{\sin{{2}}}{t}$$
is equivalent to system
$$\displaystyle{f{{\left({t},{x},{x}',{x}{''},{x}{''}\right)}}}={e}^{{{t}}}{\sin{{2}}}{t}-{x}-{3}{x}'$$
Hense the substitutions $$\displaystyle{x}_{{1}}={x},{x}_{{2}}={x}'={x}'_{{1}},{x}_{{3}}={x}{''}={x}'_{{2}},{x}_{{4}}={x}^{{{\left({3}\right)}}}={x}'_{{3}}$$
yield the system
$$\displaystyle{x}'_{{1}}={x}_{{2}}$$
$$\displaystyle{x}'_{{2}}={x}_{{3}}$$
$$x'_3=x_4$$
$$\displaystyle{x}'_{{4}}={e}^{{{t}}}{\sin{{2}}}{t}-{x}_{{1}}-{3}{x}_{{3}}$$ is a system of first-order equation.