2.2E: Exact Equations (Exercises)

In Exercises 1-19 determine which equations are exact and, if they are exact, solve them.

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

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

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

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

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

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

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

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

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

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

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

19. \((1+\ln x+{y\over x})dx=(1-\ln x)dy\)

In Exercises 20-25 solve the initial value problem.

20. \((4x^3y^2-6x^2y-2x-3)+(2x^4y-2x^3)\,{dy\over dx}=0,\quad y(1)=3\)

21. \((-4y\cos x+4\sin x\cos x+\sec^2x)\,dx=(4\sin x-4y)\,dy,\quad y(\pi/4)=0\)

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

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

24. \((2x-1)(y-1)\,dx+(x+2)(x-3)\,dy=0,\quad y(1)=-1\)

25. \({dy \over dx}={{xy^2-\cos x \sin x}\over y-yx^2},\quad y(0)=2\)

26. Find all functions \(M\) such that the equation is exact.

1. \(M(x,y)\,dx+(x^2-y^2)\,dy=0\)
2. \(M(x,y)\,dx+2xy\sin x\cos y\,dy=0\)
3. \(M(x,y)\,dx+(e^x-e^y\sin x)\,dy=0\)

27. Find all functions \(N\) such that the equation is exact.

1. \((x^3y^2+2xy+3y^2)\,dx+N(x,y)\,dy=0\)
2. \((\ln xy+2y\sin x)\,dx+N(x,y)\,dy=0\)
3. \((x\sin x+y\sin y)\,dx+N(x,y)\,dy=0\)

28. Prove: If the equations \(M_1\,dx+N_1\,dy=0\) and \(M_2\, dx+N_2\,dy=0\) are exact on an open rectangle \(R\), so is the equation \[(M_1+M_2)\,dx+(N_1+N_2)\,dy=0.\nonumber \]

29. Find conditions on the constants \(A\), \(B\), \(C\), and \(D\) such that the equation \[(Ax+By)\,dx+(Cx+Dy)\,dy=0\nonumber \] is exact.

30. Find conditions on the constants \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) such that the equation \[(Ax^2+Bxy+Cy^2)\,dx+(Dx^2+Exy+Fy^2)\,dy=0\nonumber \] is exact.

31. Rewrite the separable equation \[h(y)y'=g(x) \tag{A} \] as an exact equation \[M(x,y)\,dx+N(x,y)\,dy=0. \tag{B} \] Show that applying the method of this section to (B) yields the same solutions that would be obtained by applying the method of separation of variables to (A)