y'=cos² x cos y

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y&#039;=cos² x cos y

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To solve the differential equation dydx=cos2(x)cos(y), we can separate the variables and integrate both sides.Using the separation of variables technique, we can rewrite the equation as:1cos2(y)dy=cos2(x)dxNow, let&#039;s integrate both sides:∫1cos2(y)dy=∫cos2(x)dxOn the left-hand side, we can apply the identity sec2(y)=1+tan2(y), which gives us:∫sec2(y)dy=∫cos2(x)dxIntegrating both sides:tan(y)=∫cos2(x)dxTo find the integral of cos2(x), we can use the trigonometric identity cos2(x)=12(1+cos(2x)). Substituting this into the equation, we have:tan(y)=∫12(1+cos(2x))dxExpanding the integral:tan(y)=12∫(1+cos(2x))dxIntegrating each term separately:tan(y)=12(∫1dx+∫cos(2x)dx)The integral of 1 with respect to x is simply x. For the integral of cos(2x), we can use the substitution u=2x, which gives us du=2dx. Therefore, the integral becomes:tan(y)=12(x+∫cos(u)du2)Simplifying further:tan(y)=12(x+12∫cos(u)du)Now, integrating cos(u):tan(y)=12(x+12sin(u))Substituting back u=2x:tan(y)=12(x+12sin(2x))Therefore, the general solution to the differential equation dydx=cos2(x)cos(y) is given by:tan(y)=12(x+12sin(2x))

y'=cos² x cos y

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