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Q. 63

Expert-verifiedFound in: Page 945

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Solve the exact differential equations in Exercises 63–66. ${e}^{y}+(x{e}^{y}-7)\frac{dy}{dx}=0$

The solution of given exact differential equation is: $x{e}^{y}-7y+C=0$

Exact differential equation, ${e}^{y}+(x{e}^{y}-7)\frac{dy}{dx}=0$

$\mathrm{Given},{e}^{y}+(x{e}^{y}-7)\frac{dy}{dx}=0\phantom{\rule{0ex}{0ex}}\Rightarrow {e}^{y}dx+(x{e}^{y}-7)dy=0\phantom{\rule{0ex}{0ex}}\mathrm{It}\mathrm{is}\mathrm{a}\mathrm{exact}\mathrm{differential}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{the}\mathrm{form}Mdx+Ndy=0,\phantom{\rule{0ex}{0ex}}\mathrm{with}\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.\mathrm{the}\mathrm{solution}\mathrm{is}\mathrm{given}\mathrm{by},\phantom{\rule{0ex}{0ex}}f(x,y)=\int (\mathrm{treating}yas\mathrm{constant}\mathrm{in}\mathrm{M})dx+\int (\mathrm{terms}\mathrm{independent}\mathrm{of}x\mathrm{in}N)dy=0\phantom{\rule{0ex}{0ex}}\Rightarrow \int {e}^{y}dx+\int (-7)dy=0\phantom{\rule{0ex}{0ex}}\Rightarrow x{e}^{y}-7y+C=0$

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