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2.2E: Exact Equations (Exercises)

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    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)

    This page titled 2.2E: Exact Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.