3.7E: Excercises

Exercise $\PageIndex{1}$

In Exercises 1-17 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. $(3x^2\cos xy-x^3y\sin xy+4x)\,dx+(8y-x^4\sin xy)\,dy=0$

Exercise $\PageIndex{2}$

In Exercises 18-22 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. $(2x-1)(y-1)\,dx+(x+2)(x-3)\,dy=0,\quad y(1)=-1$

Exercise $\PageIndex{3}$

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 3.5.37(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 3.5.37(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

$$y'+2xy=-e^{-x^2}\left({3x+2ye^{x^2}\over2x+3ye^{x^2}}\right),\quad y(0)=-1.\nonumber$$

41. 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)

42. Suppose all second partial derivatives of $M=M(x,y)$ and $N=N(x,y)$ are continuous and $M\,dx+N\,dy=0$ and $-N\,dx+M\,dy=0$ are exact on an open rectangle $R$. Show that $M_{xx}+M_{yy}=N_{xx}+N_{yy}=0$ on $R$.

43. Suppose all second partial derivatives of $F=F(x,y)$ are continuous and $F_{xx}+F_{yy}=0$ on an open rectangle $R$. (A function with these properties is said to be harmonic; see also Exercise 2.5.42.) Show that $-F_y\,dx+F_x\,dy=0$ is exact on $R$, and therefore there’s a function $G$ such that $G_x=-F_y$ and $G_y=F_x$ in $R$. (A function $G$ with this property is said to be a harmonic conjugate of $F$.)

44. Verify that the following functions are harmonic, and find all their harmonic conjugates. (See Exercise 3.5.43.)

1. $x^2-y^2$
2. $e^x\cos y$
3. $x^3-3xy^2$
4. $\cos x\cosh y$
5. $\sin x\cosh y$