`y(1+x^2)y' - x(1+y^2) = 0 , y(0) = sqrt(3)` Find the particular solution that satisfies the initial condition

Wednesday, January 30, 2013

$\displaystyle{y}{\left({1}+{{x}}^{{2}}\right)}{y}'-{x}{\left({1}+{{y}}^{{2}}\right)}={0},{y}{\left({0}\right)}=\sqrt{{{3}}}$ Find the particular solution that satisfies the initial condition

This equation is separable: $\displaystyle{x}$ and $\displaystyle{y}$ may be moved to the different sides. The equivalent equation is

$\displaystyle\frac{{{y}{y}'}}{{{1}+{{y}}^{{2}}}}=\frac{{x}}{{{1}+{{x}}^{{2}}}},$

and now we can integrate it:

$\displaystyle\int\frac{{{y}{y}'}}{{{1}+{{y}}^{{2}}}}{\left.{d}{x}\right.}=\int\frac{{x}}{{{1}+{{x}}^{{2}}}}{\left.{d}{x}\right.},$

$\displaystyle\int\frac{{{y}}}{{{1}+{{y}}^{{2}}}}{\left.{d}{y}\right.}=\int\frac{{x}}{{{1}+{{x}}^{{2}}}}{\left.{d}{x}\right.},$

$\displaystyle\frac{{1}}{{2}}{\ln{{\left({1}+{{y}}^{{2}}\right)}}}=\frac{{1}}{{2}}{\ln{{\left({1}+{{x}}^{{2}}\right)}}}+{C},$

$\displaystyle{\ln{{\left({1}+{{y}}^{{2}}\right)}}}={\ln{{\left({1}+{{x}}^{{2}}\right)}}}+{C},$ (1)

where $\displaystyle{C}$ is an arbitrary constant.

If $\displaystyle{y}=\sqrt{{{3}}}$ for $\displaystyle{x}={0},$ then $\displaystyle{\ln{{\left({1}+{3}\right)}}}={\ln{{\left({1}\right)}}}+{C},$ so $\displaystyle{C}={\ln{{4}}}-{\ln{{1}}}={\ln{{4}}}.$

Finally, apply exponent to the both sides of (1) and obtain

$\displaystyle{1}+{{y}}^{{2}}={{e}}^{{{\ln{{4}}}}}{\left({1}+{{x}}^{{2}}\right)}={4}+{4}{{x}}^{{2}},$ or $\displaystyle{y}=\sqrt{{{3}+{4}{{x}}^{{2}}}}.$ This is the answer.

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Wednesday, January 30, 2013

$\displaystyle{y}{\left({1}+{{x}}^{{2}}\right)}{y}'-{x}{\left({1}+{{y}}^{{2}}\right)}={0},{y}{\left({0}\right)}=\sqrt{{{3}}}$ Find the particular solution that satisfies the initial condition

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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, January 30, 2013

Find the particular solution that satisfies the initial condition

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?

$\displaystyle{y}{\left({1}+{{x}}^{{2}}\right)}{y}'-{x}{\left({1}+{{y}}^{{2}}\right)}={0},{y}{\left({0}\right)}=\sqrt{{{3}}}$ Find the particular solution that satisfies the initial condition

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