(a) Consider the differential equation (DE) y0 = f(x, y) where f(x, y) = squaeroot(2y +x^2) (i) Without solving the DE, plot direction fields where y0 = 0 and y0 = 2. Explain whether it is possible or not to sketch a solution curve above and below curves associated to y 0 = 0. Sketch a graph of any solution curve that satisfies the DE. (ii) For what choices of (x, y) will the the initial value problem y0 =p2y + x2 have a unique solution on an open interval (a, b) that contains x? (b) Given the differential equation (DE) (y^2cos x − kx^2y − 2x)dx + (2y sin x − x^k + ln y)dy = 0. (i) Find the value of k so that the given differential equation is exact. (ii) Using your answer in (i), use the method for exact equations to solve the DE with initial condition y(0) = e.
(a) Consider the differential equation (DE) y0 = f(x, y) where f(x, y) = squaeroot(2y +x^2)
(i) Without solving the DE, plot direction fields where y0 = 0 and y0 = 2. Explain whether it is possible or not to sketch a solution curve above and below curves associated to y 0 = 0. Sketch a graph of any solution curve that satisfies the DE.
(ii) For what choices of (x, y) will the the initial value problem y0 =p2y + x2 have a unique solution on an open interval (a, b) that contains x?
(b) Given the differential equation (DE)
(y^2cos x − kx^2y − 2x)dx + (2y sin x − x^k + ln y)dy = 0.
(i) Find the value of k so that the given differential equation is exact.
(ii) Using your answer in (i), use the method for exact equations to solve the DE with initial condition y(0) = e.
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