Question

Solve differential equation x\frac{dy}{dx}-2y=x^3 \sin^2(x)

First order differential equations
ANSWERED
asked 2020-11-27
Solve differential equation \(\displaystyle{x}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}-{2}{y}={x}^{{3}}{{\sin}^{{2}}{\left({x}\right)}}\)

Answers (1)

2020-11-28
\(\displaystyle{x}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}-{2}{y}={x}^{{3}}{{\sin}^{{2}}{\left({x}\right)}}\)
\(\displaystyle{x}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}-{\frac{{{2}}}{{{x}}}}={x}^{{2}}{\sin{{\left({x}\right)}}}\)
\(\displaystyle{I}.{F}.={e}^{{-\int{\frac{{{2}}}{{{x}}}}{\left.{d}{x}\right.}}}\)
\(\displaystyle={e}^{{-{2}{\ln{{x}}}}}\)
\(\displaystyle={e}{\left\lbrace{\ln{{x}}}^{{-{2}}}\right\rbrace}\)
\(\displaystyle={x}^{{-{2}}}\)
\(\displaystyle{y}{x}^{{-{2}}}=\int{x}^{{-{2}}}{\left({x}^{{2}}{{\sin}^{{2}}{x}}\right)}{\left.{d}{x}\right.}+{c}\)
\(\displaystyle=\int{{\sin}^{{2}}{x}}{\left.{d}{x}\right.}+{c}\)
\(\displaystyle=\int{\frac{{{1}-{\cos{{2}}}{x}}}{{{2}}}}{\left.{d}{x}\right.}+{c}\)
\(\displaystyle={\frac{{{x}}}{{{2}}}}-{\frac{{{\sin{{2}}}{x}}}{{{4}}}}+{c}\)
\(\displaystyle{y}={\frac{{{x}^{{3}}}}{{{2}}}}-{\frac{{{x}^{{2}}{\sin{{2}}}{x}}}{{{4}}}}+{c}{x}^{{2}}\)
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