2.5E: Exact Equations (Exercises)

Q2.5.1

In Exercises 2.5.1-2.5.17 determine which equations are exact and solve them.

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. \((-2y^2\sin x+3y^3-2x)\,dx+(4y\cos x+9xy^2)\,dy=0\)

8. \((2x+y)\,dx+(2y+2x)\,dy=0\)

9. \((3x^2+2xy+4y^2)\,dx+(x^2+8xy+18y)\,dy=0\)

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

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

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

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

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

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

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

17. \((3x^2\cos xy-x^3y\sin xy+4x)\,dx+(8y-x^4\sin xy)\,dy=0\)

Q2.5.2

In Exercises 2.5.18-2.5.22 solve the initial value problem.

18. \((4x^3y^2-6x^2y-2x-3)\,dx+(2x^4y-2x^3)\,dy=0,\quad y(1)=3\)

19. \((-4y\cos x+4\sin x\cos x+\sec^2x)\,dx+ (4y-4\sin x)\,dy=0,\quad y(\pi/4)=0\)

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

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

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

Q2.5.3

23. Solve the exact equation \[(7x+4y)\,dx+(4x+3y)\,dy=0.\nonumber \] Plot a direction field and some integral curves for this equation on the rectangle \[\{-1\le x\le1,-1\le y\le1\}.\nonumber \]

24. Solve the exact equation \[e^x(x^4y^2+4x^3y^2+1)\,dx+(2x^4ye^x+2y)\,dy=0.\nonumber \] Plot a direction field and some integral curves for this equation on the rectangle \[\{-2\le x\le2,-1\le y\le1\}.\nonumber \]

25. Plot a direction field and some integral curves for the exact equation \[(x^3y^4+x)\,dx+(x^4y^3+y)\,dy=0\nonumber \] on the rectangle \(\{-1\le x\le 1,-1\le y\le1\}\). (See Exercise 2.5.37(a)).

26. Plot a direction field and some integral curves for the exact equation \[(3x^2+2y)\,dx+(2y+2x)\,dy=0\nonumber \] on the rectangle \(\{-2\le x\le 2,-2\le y\le2\}\). (See Exercise 2.5.37(b)).

27.

1. Solve the exact equation \[(x^3y^4+2x)\,dx+(x^4y^3+3y)\,dy=0 \tag{A} \] implicitly.
2. For what choices of \((x_0,y_0)\) does Theorem 2.3.1 imply that the initial value problem \[(x^3y^4+2x)\,dx+(x^4y^3+3y)\,dy=0,\quad y(x_0)=y_0, \tag{B}\] has a unique solution on an open interval \((a,b)\) that contains \(x_0\)?
3. Plot a direction field and some integral curves for (A) on a rectangular region centered at the origin. What is the interval of validity of the solution of (B)?

28.

1. Solve the exact equation \[(x^2+y^2)\,dx+2xy\,dy=0 \tag{A} \] implicitly.
2. For what choices of \((x_0,y_0)\) does Theorem 2.3.1 imply that the initial value problem \[(x^2+y^2)\,dx+2xy\,dy=0,\quad y(x_0)=y_0, \tag{B} \] has a unique solution \(y=y(x)\) on some open interval \((a,b)\) that contains \(x_0\)?
3. Plot a direction field and some integral curves for (A). From the plot determine, the interval \((a,b)\) of b, the monotonicity properties (if any) of the solution of (B), and \(\lim_{x\to a+}y(x)\) and \(\lim_{x\to b-}y(x)\).

29. 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. \(M(x,y)\,dx+(e^x-e^y\sin x)\,dy=0\)

30. 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. \((x\sin x+y\sin y)\,dx+N(x,y)\,dy=0\)

31. Suppose \(M,N,\) and their partial derivatives are continuous on an open rectangle \(R\), and \(G\) is an antiderivative of \(M\) with respect to \(x\); that is, \[{\partial G\over\partial x}=M.\nonumber \] Show that if \(M_y\ne N_x\) in \(R\) then the function \[N-{\partial G\over\partial y}\nonumber \] is not independent of \(x\).

32. 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 \[(M_1+M_2)\,dx+(N_1+N_2)\,dy=0.\nonumber \]

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

34. 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.

35. Suppose \(M\) and \(N\) are continuous and have continuous partial derivatives \(M_y\) and \(N_x\) that satisfy the exactness condition \(M_y=N_x\) on an open rectangle \(R\). Show that if \((x,y)\) is in \(R\) and \[F(x,y)=\int^x_{x_0}M(s,y_0)\,ds+\int^y_{y_0}N(x,t)\,dt,\nonumber \] then \(F_x=M\) and \(F_y=N\).

36. Under the assumptions of Exercise 2.5.35, show that \[F(x,y)=\int^y_{y_0}N(x_0,s)\,ds+\int^x_{x_0}M(t,y)\,dt.\nonumber \]

37. Use the method suggested by Exercise 2.5.35, with \((x_0,y_0)=(0,0)\), to solve the these exact equations:

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

38. Solve the initial value problem \[y'+{2\over x}y=-{2xy\over x^2+2x^2y+1},\quad y(1)=-2.\nonumber \]

39. Solve the initial value problem \[y'-{3\over x}y={2x^4(4x^3-3y)\over3x^5+3x^3+2y},\quad y(1)=1.\nonumber \]

40. Solve the initial value problem \[y'+2xy=-e^{-x^2}\left({3x+2ye^{x^2}\over2x+3ye^{x^2}}\right),\quad y(0)=-1.\nonumber \]