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Mathematics LibreTexts

2.5E: Exact Equations (Exercises)

  • Page ID
    18245
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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. \(\sin x\cosh y\)