DE1summer07

# DE1summer07 - MTH 2201 Differential Equations Homework 1...

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Unformatted text preview: MTH 2201 Differential Equations Homework 1 Spring 2008 1. Verify that y(t) = e-t/2 is a solution of the DE 2y + y = 0. 2. Verify that x2 y +y 2 = 1 is a solution of the DE 2xydx+(x2 +2y)dy = 0. (Assume an appropriate interval of definition.) 3. Let y(t) = e-t 2 t 0 eu du + c1 e-t , where c1 is an arbitrary constant. 2 2 Show that y(t) is a solution of y + 2ty = 1. 4. Verify that x(t) = e-2t + 3e6t , y(t) = -e-2t + 5e6t is a solution of the system of DE on (-, ) dx dy = x + 3y = 5x + 3y. dt dt 5. Make up a DE that doesnot have any real solutions. 6. Consider t(y )2 -4y -12t3 = 0. Can we write it in the form 7. Verify that y(t) = e- sin t is a solution of y + (cos t)y = 0. 8. Verify that y(t) = e2it is a solution of y + 4y = 0. dy 9. Make up a DE of the form = 2y - t + g(y), that has y(t) = e2t as a dt solution. 10. Solve the ODE y = y. Describe in graphical terms the difference between the solutions of the IVPs y = y, y(0) = y0 and y = y, y(t0 ) = y0 , y0 = 0. 11. Consider the ODE y -y = 0. Let (t) be its solution on (-, ), such that (0) = 1. Verify that (t1 +t), for some real number t1 , is a solution of y - y = 0, y(0) = (t1 ). Use this and show (t1 + t) = (t1 )(t). 12. Solve the following ODEs by separation of variables : (y + 1)2 dt = . (i) dt - t2 dy = 0 (ii) y ln t dy t dP (iii) sec2 tdy + csc ydt = 0 (iv) = P - P 2. dt dy xy + 3x - y - 3 (v) = dx xy - 2x + 4y - 8 1 dy = f (t, y). dt 13. Solve the following IVPs dy (i) x2 = y - xy, y(-1) = -1. dx (ii) y = y, y(t0 ) = y0 (> 0). What can you say if y0 < 0. 14. Solve the homogeneous ODEs (i) (t - y)dt + tdy = 0. (iii) (y 2 + ty)dt - t2 dy = 0 15. Sketch in the xy - plane, the regions for which the ODE dy (1 - x2 )( )2 = 1 - y 2 has real solutions. Find solutions of the ODE dx in these regions. Find singular solutions of the equation. (ii) dy y-x = . dx y+x 2 ...
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DE1summer07 - MTH 2201 Differential Equations Homework 1...

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