Answer the following question. Follow the format given and write legibly to avoid confusions of answer. THANK YOU VERY MUCH!!

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Find the general solution of the given differential equation, and use it to determine how solutions behave as t - co. + = 3 cos(4t), t > 0 t. NOTE: Use c for the constant of integration. y Solutions converge to the function y =Solve the initial value problem. 2 ,y(0)=a Let an be the value of a for which the transition from one type of the By'8y=e long-run behavior to another occurs. Find the critical value a.) exactly. Nx - My If XM - WN = R, where R is a function depending only on the quantity z = xy, then the differential equation M + Ny' = 0 has an integrating factor of the form R(z)dz H(z) = e. Find an integrating factor and solve the given equation. 672 18* + 3 18v dy + =0 X dx O 1 u(xy) = - 6x2 v x 3x2 xy 2 + 6y = c O 1 3.x2 H(xy) = - 6x y + - + 6y = c xy 2 O 3x2 u(xy) = xy, 6xy+ 2 - + 6y = C O 3x3 H(xy) = - 6x y + + 6V = c xy 2 O 3.x3 u(xy) = xy, 6xy+ * + by = cNewton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its Surroundings. Suppose that the temperature of a cup of coffee obeye Newton's law of cooling. If the coffee has a temperature of 184 \"F when freshly poured, and 1 min later has cooled to 174 I3F in a room at 70 \"F... determine when the coffee reaches a temperature of 138 \"F. Note: Enter the exact symbolic expression for the asked time- The coffee reaches the temperature 138 in