a) Solve the given differential equation by separation of variables. 𝑠𝑒𝑐 2 𝑥𝑑𝑦 + 𝑐𝑠𝑐 𝑦𝑑𝑥 = 0

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To solve the given differential equation sec2(x)dy+csc(y)dx=0 using separation of variables, we can rearrange the terms and integrate both sides.Let&#039;s begin by isolating the variables. We have:sec2(x)dy=−csc(y)dxNext, we can separate the variables by dividing both sides:dycsc(y)=−dxsec2(x)To simplify the expression on the left-hand side, we can rewrite csc(y) as 1sin(y) and on the right-hand side, we can rewrite sec2(x) as 1cos2(x). This gives us:dy1sin(y)=−dx1cos2(x)Simplifying further:sin(y)dy=−cos2(x)dxNow, let&#039;s integrate both sides:∫sin(y)dy=−∫cos2(x)dxTo integrate sin(y) with respect to y, we can use the substitution u=cos(y), which implies du=−sin(y)dy. Therefore, the integral becomes:−∫du=−∫cos2(x)dxSimplifying further:−u=−∫cos2(x)dxUsing the trigonometric identity cos2(x)=12(1+cos(2x)), we have:−u=−∫12(1+cos(2x))dxExpanding the integral:−u=−12∫(1+cos(2x))dxIntegrating each term separately:−u=−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:−u=−12(x+∫cos(u)du2)Simplifying further:−u=−12(x+12∫cos(u)du)Now, integrating cos(u):−u=−12(x+12sin(u))Substituting back u=2x:−u=−12(x+12sin(2x))Therefore, the general solution to the given differential equation sec2(x)dy+csc(y)dx=0 is:−cos(y)=−12(x+12sin(2x))Simplifying further, we have:cos(y)=12(x+12sin(2x))Thus, we have found the solution to the given differential equation.

a) Solve the given differential equation by separation

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