APM 2611- DIFFERENTIAL EQUATIONS-2019

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Question 1 (1.1) Find an interval of definition for the equation x2 dx = 3x3t2. dt Since the equation has the solution x(t) = 0 for all t, any interval may be taken as an interval of definiti... on. The largest interval of definition is (−∞, ∞). (1.2) Solve the initial value problem x2 dx = 3x3t2, x(0) = 1 dt and give an interval of definition. Notice that x = 0 is a solution for the equation, but does not satisfy the initial value problem. The equation is separable, separating variables yields dx = 3t2 dt. x Integrating both sides yields ln |x| = t3 + c where c is the constant of integration. Since x(0) = 1 ln |1| = 03 + c we have c = 0. Thus ln |x| = t3 ⇒ |x| = et3 . Since x(0) = 1 (which is positive), and by continuity of x(t) x = t3 Since the solution is valid for all t, an interval of definition is ( , ) or any other interval which contains the initial value 0. CONTINUED..... [Show More]

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