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2.5E: Exact Equations (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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

[exer:2.5.1] $$6x^2y^2\,dx+4x^3y\,dy=0$$

[exer:2.5.2] $$(3y\cos x+4xe^x+2x^2e^x)\,dx+(3\sin x+3)\,dy=0$$

[exer:2.5.3] $$14x^2y^3\,dx+21 x^2y^2\,dy=0$$

[exer:2.5.4] $$(2x-2y^2)\,dx+(12y^2-4xy)\,dy=0$$

[exer:2.5.5]$$(x+y)^2\,dx+(x+y)^2\,dy=0$$

[exer:2.5.6]$$(4x+7y)\,dx+(3x+4y)\,dy=0$$

[exer:2.5.7] $$(-2y^2\sin x+3y^3-2x)\,dx+(4y\cos x+9xy^2)\,dy=0$$

[exer:2.5.8] $$(2x+y)\,dx+(2y+2x)\,dy=0$$

[exer:2.5.9] $$(3x^2+2xy+4y^2)\,dx+(x^2+8xy+18y)\,dy=0$$

[exer:2.5.10] $$(2x^2+8xy+y^2)\,dx+(2x^2+xy^3/3)\,dy=0$$

[exer:2.5.11] $${\left({1\over x}+2x\right)\,dx+ \left({1\over y}+2y\right)\,dy=0}$$

[exer:2.5.12] $$(y\sin xy+xy^2\cos xy)\,dx+(x\sin xy+xy^2\cos xy)\,dy=0$$

[exer:2.5.13] $${{x\,dx\over(x^2+y^2)^{3/2}}+{y\,dy \over(x^2+y^2)^{3/2}}=0}$$

[exer:2.5.14] $$\left(e^x(x^2y^2+2xy^2)+6x\right)\,dx+(2x^2ye^x+2)\,dy=0$$

[exer:2.5.15] $$\left(x^2e^{x^2+y}(2x^2+3)+4x\right)\,dx+(x^3e^{x^2+y}-12y^2)\,dy=0$$

[exer:2.5.16] $$\left(e^{xy}(x^4y+4x^3)+3y\right)\,dx+(x^5e^{xy}+3x)\,dy=0$$

[exer:2.5.17] $$(3x^2\cos xy-x^3y\sin xy+4x)\,dx+(8y-x^4\sin xy)\,dy=0$$

In Exercises 18–22 solve the initial value problem.

[exer:2.5.18] $$(4x^3y^2-6x^2y-2x-3)\,dx+(2x^4y-2x^3)\,dy=0,\quad y(1)=3$$

[exer:2.5.19] $$(-4y\cos x+4\sin x\cos x+\sec^2x)\,dx+ (4y-4\sin x)\,dy=0,\quad y(\pi/4)=0$$

[exer:2.5.20] $$(y^3-1)e^x\,dx+3y^2(e^x+1)\,dy=0,\quad y(0)=0$$

[exer:2.5.21] $$(\sin x-y\sin x-2\cos x)\,dx+\cos x\,dy=0,\quad y(0)=1$$

[exer:2.5.22] $$(2x-1)(y-1)\,dx+(x+2)(x-3)\,dy=0,\quad y(1)=-1$$

[exer:2.5.23] 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 $\{-1\le x\le1,-1\le y\le1\}.\nonumber$

[exer:2.5.24] 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 $\{-2\le x\le2,-1\le y\le1\}.\nonumber$

[exer:2.5.25] 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 [exer:2.5.37] a).

[exer:2.5.26] 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 [exer:2.5.37](b)).

[exer:2.5.27]

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 [thmtype: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)?

[exer:2.5.28]

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 [thmtype: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 $$\lim_{x\to b-}y(x)$$.

[exer:2.5.29] 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. $$M(x,y)\,dx+(e^x-e^y\sin x)\,dy=0$$

[exer:2.5.30] 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. $$(x\sin x+y\sin y)\,dx+N(x,y)\,dy=0$$

[exer:2.5.31] 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 $$x$$.

[exer:2.5.32] 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 $(M_1+M_2)\,dx+(N_1+N_2)\,dy=0.\nonumber$

[exer:2.5.33] Find conditions on the constants $$A$$, $$B$$, $$C$$, and $$D$$ such that the equation $(Ax+By)\,dx+(Cx+Dy)\,dy=0\nonumber$ is exact.

[exer:2.5.34] 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.

[exer:2.5.35] 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 $$F_y=N$$.

[exer:2.5.36] Under the assumptions of Exercise [exer:2.5.35], show that $F(x,y)=\int^y_{y_0}N(x_0,s)\,ds+\int^x_{x_0}M(t,y)\,dt.\nonumber$

[exer:2.5.37] Use the method suggested by Exercise [exer: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. $$(3x^2+2y)\,dx+(2y+2x)\,dy=0$$

[exer:2.5.38] Solve the initial value problem $y'+{2\over x}y=-{2xy\over x^2+2x^2y+1},\quad y(1)=-2.\nonumber$

[exer:2.5.39] Solve the initial value problem $y'-{3\over x}y={2x^4(4x^3-3y)\over3x^5+3x^3+2y},\quad y(1)=1.\nonumber$

[exer:2.5.40] Solve the initial value problem $y'+2xy=-e^{-x^2}\left({3x+2ye^{x^2}\over2x+3ye^{x^2}}\right),\quad y(0)=-1.\nonumber$

[exer:2.5.41] Rewrite the separable equation $h(y)y'=g(x) \tag{\rm (A)}\nonumber$ as an exact equation $M(x,y)\,dx+N(x,y)\,dy=0. \tag{\rm (B)}\nonumber$ 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)

[exer:2.5.42] 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 $$R$$.

[exer:2.5.43] 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 [exer: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 $$F$$.)

[exer:2.5.44] Verify that the following functions are harmonic, and find all their harmonic conjugates. (See Exercise [exer:2.5.43].)

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