`y = 3e^(2x) - 4sin(2x)` Determine whether the function is a solution of the differential equation `y^((4))

Given,

$\displaystyle{y}={3}{{e}}^{{{2}{x}}}-{4}{\sin{{\left({2}{x}\right)}}}$

so,

we have to find

$\displaystyle{y}'={\left({3}{{e}}^{{{2}{x}}}-{4}{\sin{{\left({2}{x}\right)}}}\right)}'={\left({3}{{e}}^{{{2}{x}}}\right)}'-{\left({4}{\sin{{\left({2}{x}\right)}}}\right)}'$

$\displaystyle={3}\cdot{2}{{e}}^{{{2}{x}}}-{2}\cdot{4}{\cos{{\left({2}{x}\right)}}}$

$\displaystyle={6}{{e}}^{{{2}{x}}}-{8}{\cos{{\left({2}{x}\right)}}}$

similarly

$\displaystyle{y}''={\left({6}{{e}}^{{{2}{x}}}-{8}{\cos{{\left({2}{x}\right)}}}\right)}'$

$\displaystyle={6}\cdot{2}{{e}}^{{{2}{x}}}+{2}\cdot{8}{\sin{{\left({2}{x}\right)}}}$

$\displaystyle={12}{{e}}^{{{2}{x}}}+{16}{\sin{{\left({2}{x}\right)}}}$

$\displaystyle{y}'''={\left({12}{{e}}^{{{2}{x}}}+{16}{\sin{{\left({2}{x}\right)}}}\right)}'$

$\displaystyle={12}\cdot{2}{{e}}^{{{2}{x}}}+{16}\cdot{2}{\cos{{\left({2}{x}\right)}}}$

$\displaystyle={24}{{e}}^{{{2}{x}}}+{32}{\cos{{\left({2}{x}\right)}}}$

$\displaystyle{y}''''={\left({24}{{e}}^{{{2}{x}}}+{32}{\cos{{\left({2}{x}\right)}}}\right)}'$

$\displaystyle={24}\cdot{2}{{e}}^{{{2}{x}}}-{32}\cdot{2}{\sin{{\left({2}{x}\right)}}}$

$\displaystyle={48}{{e}}^{{{2}{x}}}-{64}{\sin{{\left({2}{x}\right)}}}$

So lets see whether $\displaystyle{y}''''-{16}{y}={0}$

=> $\displaystyle{48}{{e}}^{{{2}{x}}}-{64}{\sin{{\left({2}{x}\right)}}}-{16}{\left({3}{{e}}^{{{2}{x}}}-{4}{\sin{{\left({2}{x}\right)}}}\right)}$

$\displaystyle={48}{{e}}^{{{2}{x}}}-{64}{\sin{{\left({2}{x}\right)}}}-{48}{{e}}^{{{2}{x}}}+{64}{\sin{{\left({2}{x}\right)}}}={0}$

so,

$\displaystyle{y}''''-{16}{y}={0}$

Blog

Sunday, February 22, 2015

$\displaystyle{y}={3}{{e}}^{{{2}{x}}}-{4}{\sin{{\left({2}{x}\right)}}}$ Determine whether the function is a solution of the differential equation $\displaystyle{{y}}^{{{\left({4}\right)}}}-{16}{y}={0}$

No comments:

Post a Comment

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Sunday, February 22, 2015

Determine whether the function is a solution of the differential equation

No comments:

Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{y}={3}{{e}}^{{{2}{x}}}-{4}{\sin{{\left({2}{x}\right)}}}$ Determine whether the function is a solution of the differential equation $\displaystyle{{y}}^{{{\left({4}\right)}}}-{16}{y}={0}$

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?