`dy/dx = xcos(x^2)` Use integration to find a general solution to this differential equation.

Saturday, January 5, 2013

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={x}{\cos{{\left({{x}}^{{2}}\right)}}}$ Use integration to find a general solution to this differential equation.

To solve this differential equation, rewrite it as

$\displaystyle{\left.{d}{y}\right.}={x}{\cos{{\left({{x}}^{{2}}\right)}}}{\left.{d}{x}\right.}$

Integrate both sides of the equation:

$\displaystyle{y}=\int{x}{\cos{{\left({{x}}^{{2}}\right)}}}{\left.{d}{x}\right.}$

To take the integral on the right side of the equation, use the substitution method. Let $\displaystyle{z}{\left({x}\right)}={{x}}^{{2}}$ .

Then, $\displaystyle{\left.{d}{z}\right.}={2}{x}{\left.{d}{x}\right.}$ and integral becomes

$\displaystyle\int{x}{\cos{{\left({{x}}^{{2}}\right)}}}{\left.{d}{x}\right.}=\int{\left({x}{\left.{d}{x}\right.}\right)}{\cos{{\left({{x}}^{{2}}\right)}}}=\int\frac{{1}}{{2}}{\left.{d}{z}\right.}{\cos{{z}}}=\frac{{1}}{{2}}\int{\cos{{z}}}{\left.{d}{z}\right.}$

This is a simple trigonometric integral: $\displaystyle\int{\cos{{z}}}{\left.{d}{z}\right.}={\sin{{z}}}$ . Substituting the original variable, x, back into equation results in

$\displaystyle{y}=\frac{{1}}{{2}}{{\sin{{x}}}}^{{2}}+{C}$ , where C is an arbitrary constant.

So, the general solution of the given differential equation is

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

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Saturday, January 5, 2013

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={x}{\cos{{\left({{x}}^{{2}}\right)}}}$ Use integration to find a general solution to this 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?

Saturday, January 5, 2013

Use integration to find a general solution to this 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?

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={x}{\cos{{\left({{x}}^{{2}}\right)}}}$ Use integration to find a general solution to this differential equation.

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