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Department of MathematicsMAL 110 (Mathematics I)

Tutorial Sheet No. 4Differential Equations of First Order

1. Find all solutions of the following equation

(a) y + 2xy = x

(b) xy + y = 3x3 1

(c) y + 2xy = xex2

(d) y = exy

1+ex

2. Consider the equation:y + (cosx)y = esinx

Find the solution which satisfies () =

3. Consider the equation x2y + 2xy = 1 ; 0 < x <

(a) Show that every solution tends to zero as x (b) Find every solution which satisfies(2 ) = 2(1)

4. Solve:

(a) y = xy+2x+y1

(b) y = x+y+12x+2y1

5. Find the orthogonal trajectories for the following:

(a) ex ey =c

(b) cosx.sinhy= c

(c) y2 = 4c(c+x)(d) r= c(sec + tan)

6. The equation below is written in the form M(x, y)dx + N(x, y)dy = 0, where M, N exist on the wholeplane.

(a) Determine whether the following is exact or not2xydx + (x2 + 3y2)dy = 0

(b) Determine whether the following equation is exact or not and solveexdx + (ey(y+ 1))dy = 0

(c) Determine whether the following equation is exactcosxcos2ydx sinxsin2ydy = 0

(d) Determine whether the following is exact or notx2y3dx x3y2dy = 0

7. Consider the equation M(x, y)dx + N(x, y)dy = 0 where M, N have continuous first partial derivativeson some rectangle R, where R = (x, y) : |x x0| a, |y y0| b. Prove that a function u on R, havingcontinuous first partial derivatives, is an integrating factor if and only if,u( My

Nx ) = N

ux M

uy on R.

8. Consider the equationM(x, y)dx + N(x, y)dy = 0. If it has an integrating factoru, which is a function ofx alone, then P = 1N(

My

Nx) is continuous function ofx alone.

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9. Solve(2xy2 + y)dx + (2y3 x)dy = 0

10. Find a integrating factor of the formxpyq and solve(4xy2 + 6y)dx + (5x2y + 8x)dy = 0

11. Find an integrating factor of the form xpyq and solve(5x+ 2y+ 1)dx + (2x+y+ 1)dy = 0

12. Find the most general functionN(x, y) such that the following equation becomes exact

(a) (x3 +xy2)dx + N(x, y)dy = 0

(b) (x2y2 +xy3)dx + N(x, y) = 0

13. SolveSin dr

d + 2rcos + 1 = 0

14. Solve(x2 +y)y = 6x

15. Solvey dydx + x = x

2lnx

16. Solvee3ydx

dy + 3(xe3y +y) = 0

17. Reduce the following 2nd order differential equations to the first order and solve

(a) y + 9y = 0

(b) y +e2y.y3 = 0

18. By computing appropriate Lipschitz constants, show that the following functions satisfy Lipschitz conditionson the sets S indicated-

(a) f(x, y) = 4x2 +y2, on S: |x| 1; |y| 1

(b) f(x, y) = x2cos2y+ysin2x, on S: |x| 1; |y| <

19. Show that the function fgiven by f(x, y) = y1/2 does not satisfy a Lipschitz condition onR: |x| 1; 0 y 1

20. Compute the Picards first three successive approximations

(a) y = 1 +xy, y(0) = 1

(b) y = x+y, y(1) = 2

(c) y = 2xy+ 1, y(0) = 0

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