Question

Solve differential equation \frac{dy}{dx}-(\cot x)y= \sin^3x

First order differential equations
ANSWERED
asked 2021-01-16
Solve differential equation \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}-{\left({\cot{{x}}}\right)}{y}={{\sin}^{{3}}{x}}\)

Answers (1)

2021-01-17
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{P}{\left({n}\right)}{y}={Q}{\left({n}\right)}\)
\(\displaystyle{I}.{F}={e}^{{\int{p}{\left({x}\right)}{\left.{d}{x}\right.}}}\)
\(\displaystyle={e}^{{\int-{\cot{{x}}}{\left.{d}{x}\right.}}}\)
\(\displaystyle={e}^{{-{\ln{{\sin{{n}}}}}}}\)
\(\displaystyle={e}^{{{\ln{{\frac{{{1}}}{{{\sin{{n}}}}}}}}}}\)
\(\displaystyle{I}.{F}.={\frac{{{1}}}{{{\sin{{n}}}}}}\)
\(\displaystyle{y}{\left({I}.{F}.\right)}=\int{\left({I}.{F}.\right)}{Q}{\left({x}\right)}\)
\(\displaystyle{\frac{{{1}}}{{{\sin{{x}}}}}}{y}=\int{\left\lbrace{{\sin}^{{3}}{x}}\right\rbrace}{\left\lbrace{\sin{{x}}}\right\rbrace}{d}{n}\)
\(\displaystyle{\frac{{{y}}}{{{\sin{{x}}}}}}=\int{{\sin}^{{2}}{n}}{\left.{d}{x}\right.}\)
\(\displaystyle{\frac{{{y}}}{{{\sin{{x}}}}}}=\int{\frac{{{1}-{\cos{{2}}}{x}}}{{{2}}}}{\left.{d}{x}\right.}\)
\(\displaystyle{\frac{{{y}}}{{{\sin{{x}}}}}}=\int{\frac{{{1}}}{{{2}}}}{\left.{d}{x}\right.}-\int{\frac{{{\cos{{2}}}{x}}}{{{2}}}}{\left.{d}{x}\right.}\)
\(\displaystyle{\frac{{{y}}}{{{\sin{{x}}}}}}={\frac{{{x}}}{{{2}}}}-{\frac{{{{\sin}^{{2}}{n}}}}{{{4}}}}+{c}\)
\(\displaystyle{y}={\frac{{{2}{\sin{{x}}}{\sin{{x}}}-{\sin{{x}}}{\sin{{2}}}{x}}}{{{4}}}}+{c}\)
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