`y = x^2e^x` Determine whether this function is a solution of the differential equation `xy'

Monday, March 3, 2014

$\displaystyle{y}={{x}}^{{2}}{{e}}^{{x}}$ Determine whether this function is a solution of the differential equation $\displaystyle{x}{y}'-{2}{y}={{x}}^{{3}}{{e}}^{{x}}$ .

To determine whether the given function is a solution of the given differential equation, we can find the derivative of the function and check if it satisfies the equation.

To find the derivative of $\displaystyle{y}={{x}}^{{2}}{{e}}^{{x}}$ , use the product rule:

$\displaystyle{\left({f{{g}}}\right)}'={f{'}}{g{+}}{f{{g{'}}}}$

Here, $\displaystyle{f{=}}{{x}}^{{2}}$ and $\displaystyle{f{'}}={2}{x}$ , and $\displaystyle{g{=}}{{e}}^{{x}}$ and $\displaystyle{g{'}}={{e}}^{{x}}$ .

So $\displaystyle{y}'={2}{x}{{e}}^{{x}}+{{x}}^{{2}}{{e}}^{{x}}={x}{\left({2}+{x}\right)}{{e}}^{{x}}$ .

The left-hand side of the given equation will then be

$\displaystyle{x}{y}'-{2}{y}={2}{{x}}^{{2}}{{e}}^{{x}}+{{x}}^{{3}}{{e}}^{{x}}-{2}{{x}}^{{2}}{{e}}^{{x}}={{x}}^{{3}}{{e}}^{{x}}$ . This is exactly the same as the right-hand side of the given equation, which means $\displaystyle{y}{\left({x}\right)}={{x}}^{{2}}{{e}}^{{x}}$ is a solution.

The function $\displaystyle{y}{\left({x}\right)}={{x}}^{{2}}{{e}}^{{x}}$ is a solution of the given differential equation.

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Monday, March 3, 2014

$\displaystyle{y}={{x}}^{{2}}{{e}}^{{x}}$ Determine whether this function is a solution of the differential equation $\displaystyle{x}{y}'-{2}{y}={{x}}^{{3}}{{e}}^{{x}}$ .

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

Monday, March 3, 2014

Determine whether this function is a solution of the differential equation .

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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?

$\displaystyle{y}={{x}}^{{2}}{{e}}^{{x}}$ Determine whether this function is a solution of the differential equation $\displaystyle{x}{y}'-{2}{y}={{x}}^{{3}}{{e}}^{{x}}$ .

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