MA240_2009FALL_EXAM1__[0] - MA240 Ordinary Differential...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA240 Ordinary Differential Equations The Cooper Union, Fall 2009 Prof. Alex Casti Midterm Exam Solutions Problem 1 (20 points) Consider the following linear initial value problem: x 2 dy dx + xy = 1 y ( 1 ) = 1 (a) Solve for y ( x ) . There should be no free parameters (constants) in your final solution. Notice that the ODE can be rewritten as x dy dx + y = 1 x → d dx ( xy ) = 1 x Z x 1 d dx ( xy ) dx = Z x 1 dx x xy- y ( 1 ) = ln x ( use IC ) xy- 1 = ln x → y = 1 x ( 1 + ln x ) . (b) Describe the largest interval I on the x-axis (containing the initial data point) over which the solution exists. The solution y = x- 1 ( 1 + ln x ) is singular at x = 0, so we must exclude that point. Because we want a solution domain that contains x = 1, where the initial condition is given, we choose I = ( , ∞ ) . Problem 2 (10 points) Suppose the autonomous ODE dy dx = ye y- 9 y e y Find the critical points and classify each of them as asymptotically stable, unstable, or semi-stable. Justify your answers. The critical points correspond to solutions of ye y- 9 y e y = → y ( e y- 9 ) = . Thus, there are two critical points (critical lines) y c = 0 and y c = ln9. To assess the stability of these solutions, we investigate the sign of the derivative dy dx just below and just above the lines y = y c in the xy-plane. Consider first y c = 0. For 0 < y < ln9, we have dy dx < 0, which means that any initial condition y ( ) = y , with 0 < y < ln9, will lie on a solution curve that decreases toward the critical line at y = 0. If y < 0, we see that dy dx > 0, since dy dx = ( ye- y )( e y- 9 ) is the product of two negative terms. Therefore, any initial condition y < 0 will be on a solution curve that increases toward the critical line y = 0. From this we conclude that the critical point y c = 0 is asymptotically stable (an attractor )....
View Full Document

Page1 / 5

MA240_2009FALL_EXAM1__[0] - MA240 Ordinary Differential...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online