`y' = sqrt(x)y` Solve the differential equation

Wednesday, February 17, 2016

$\displaystyle{y}'=\sqrt{{{x}}}{y}$ Solve the differential equation

An ordinary differential equation (ODE) has differential equation for a function with single variable. A first order ODE follows $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={f{{\left({x},{y}\right)}}}$ .

It can also be in a form of $\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$ as variable separable differential equation..

To be able to set-up the problem as $\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$ , we let $\displaystyle{y}'=\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}.$

The problem: $\displaystyle{y}'=\sqrt{{{x}}}{y}$ becomes:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\sqrt{{{x}}}{y}$

Rearrange by cross-multiplication, we get:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{y}}=\sqrt{{{x}}}{\left.{d}{x}\right.}$

Apply direct integration on both sides: $\displaystyle\int\frac{{{\left.{d}{y}\right.}}}{{y}}=\int\sqrt{{{x}}}{\left.{d}{x}\right.}$ to solve for the general solution of a differential equation.

For the left side, we applying basic integration formula for logarithm:

$\displaystyle\int\frac{{{\left.{d}{y}\right.}}}{{y}}={\ln}{\left|{y}\right|}$

For the right side, we apply the Law of Exponent: $\displaystyle\sqrt{{{x}}}={{x}}^{{\frac{{1}}{{2}}}}$ then follow the Power Rule of integration: $\displaystyle\int{{x}}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}}^{{{n}+{1}}}}{{{n}+{1}}}+{C}$

$\displaystyle\int\sqrt{{{x}}}\cdot{\left.{d}{x}\right.}=\int{{x}}^{{\frac{{1}}{{2}}}}\cdot{\left.{d}{x}\right.}$

$\displaystyle=\frac{{{x}}^{{\frac{{1}}{{2}}+{1}}}}{{\frac{{1}}{{2}}+{1}}}+{C}$

$\displaystyle=\frac{{{x}}^{{\frac{{3}}{{2}}}}}{{\frac{{3}}{{2}}}}+{C}$

$\displaystyle={{x}}^{{\frac{{3}}{{2}}}}\cdot{\left(\frac{{2}}{{3}}\right)}+{C}$

$\displaystyle=\frac{{{2}{{x}}^{{\frac{{3}}{{2}}}}}}{{3}}+{C}$

Combining the results from both sides, we get the general solution of the differential equation as:

$\displaystyle{\ln}{\left|{y}\right|}=\frac{{{2}{{x}}^{{\frac{{3}}{{2}}}}}}{{3}}+{C}$

$\displaystyle{y}={{e}}^{{{\left(\frac{{{2}{{x}}^{{\frac{{3}}{{2}}}}}}{{3}}+{C}\right)}}}$

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Wednesday, February 17, 2016

$\displaystyle{y}'=\sqrt{{{x}}}{y}$ Solve the differential equation

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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?

Wednesday, February 17, 2016

Solve the differential equation

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

$\displaystyle{y}'=\sqrt{{{x}}}{y}$ Solve the differential equation

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