2.5E: Exact Equations (Exercises)

In Exercises 1-17, determine whether the differential is exact. If it is, solve it.

1. \(6x^2y^2\,dx+4x^3y\,dy=0\)

2. \((3y\cos x+4xe^x+2x^2e^x)\,dx+(3\sin x+3)\,dy=0\)

3. \(14x^2y^3\,dx+21 x^2y^2\,dy=0\)

4. \((2x-2y^2)\,dx+(12y^2-4xy)\,dy=0\)

5. \((x+y)^2\,dx+(x+y)^2\,dy=0\)

6. \((4x+7y)\,dx+(3x+4y)\,dy=0\)

7. \((2y\sin x\cos x-y+2y^2e^{xy^2})\,dx=(x-\sin^2z-4xye^{xy^2})\,dy\)

8. \((4y\cos x+9xy^2)\,dy=(2y^2\sin x-3y^3+2x)\,dx\)

9. \((2x+y)\,dx+(2y+2x)\,dy=0\)

10. \((3x^2+2xy+4y^2)\,dx+(x^2+8xy+18y)\,dy=0\)

11. \((2x^2+8xy+y^2)\,dx+(2x^2+xy^3/3)\,dy=0\)

12. \( {\left({1\over x}+2x\right)\,dx+ \left({1\over y}+2y\right)\,dy=0}\)

13. \((y\sin xy+xy^2\cos xy)\,dx+(x\sin xy+xy^2\cos xy)\,dy=0\)

14. \( {{x\,dx\over(x^2+y^2)^{3/2}}+{y\,dy \over(x^2+y^2)^{3/2}}=0}\)

15. \(\left(e^x(x^2y^2+2xy^2)+6x\right)\,dx+(2x^2ye^x+2)\,dy=0\)

16. \(\left(x^2e^{x^2+y}(2x^2+3)+4x\right)\,dx+(x^3e^{x^2+y}-12y^2)\,dy=0\)

17. \(\left(e^{xy}(x^4y+4x^3)+3y\right)\,dx+(x^5e^{xy}+3x)\,dy=0\)

18. \((3x^2\cos xy-x^3y\sin xy+4x)\,dx+(8y-x^4\sin xy)\,dy=0\)

In Exercises 19-23, solve the initial value problem.

19. \((4x^3y^2-6x^2y-2x-3)\,dx+(2x^4y-2x^3)\,dy=0,\quad y(1)=3\)

20. \((-4y\cos x+4\sin x\cos x+\sec^2x)\,dx+ (4y-4\sin x)\,dy=0, \quad y(\pi/4)=0\)

21. \((y^3-1)e^x\,dx+3y^2(e^x+1)\,dy=0,\quad y(0)=0\)

22. \((\sin x-y\sin x-2\cos x)\,dx+\cos x\,dy=0,\quad y(0)=1\)

23. \((2x-1)(y-1)\,dx+(x+2)(x-3)\,dy=0,\quad y(1)=-1\)

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