Kay's Math: Questions on exact differential equations

Determine whether or not each of the given equations is exact; solve those that are exact:

1. $y^{2}dx+ 2xy dx =0$

2. $(2x sin y + y^{3}e^{x})dx + (x^{2} cos y + 3 y^{2}e^{x})dy=0$

3. $(3x^{2}+4xy)dx + (2x^{2}+2y)dy = 0$

4. $(2x cos y + 3x^{2}y)dx + (x^{3}- x^{2}siny -y)dy = 0 ; y(0)=2$

5. $(3x +2y)dx + (2x+y)dy=0$

6. $(y^{2} + 3)dx + (2xy-4)dy=0$

7. $(2xy+1)dx+(x^{2}+4y)dy=0$

8. $(3x^{2}y+2)dx - (x^{3}+y)dy=0$

9. $(6xy+2y^{2}-5)dx + (3x^{2}+4xy-6)dy=0$

10. $(\Theta ^{2}+1)cos r dr + 2\Theta sin r d\Theta = 0$
11. $(\frac{x}{y^{2}}+x )dx + (\frac{x^{2}}{y^{3}}+y)dy=0$

12. $(\frac{2s-1}{t} )ds + (\frac{s-s^{2}}{t^{2}})dt=0$

13. $(\frac{2y^{3/2}+1}{x^{1/2}} )dx + (3x^{1/2}y^{1/2}-1)dy=0$

14. $(2xy-3)dx + (x^{2}+4y)dy=0, y(1)=2$

15. $(3x^{2}y^{2}-y^{3}+2x)dx + (2x^{3}y-3xy^{2}+1)dy=0, y(-2)=1$

16. $(2y sin x cosx+y^{2}sin x)dx + (sin^{2}x-2ycosx)dy=0, y(0)=3$

17. $(ye^{x}+2e^{x}+y^{2})dx + (e^{x}+2xy)dy=0, y(0)=6$

Kay's Math

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Sunday, January 6, 2019

Questions on exact differential equations

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