Chapter 8 Problems

Thirty chapter problems: 5 from the original book and 25 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   Implicitly derive the following functions.

   1. 
   2. 
   3. 

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2. Original problem · clarified

   Determine if the given differential equation is linear and state its order.

   1. $\mathrm{xy}″+y\prime +y^2=0$
   2. $y\prime -y=x^2$
   3. $y^{\left(5\right)}-\mathrm{yy}\prime +(x+1)y=0$

   Clarified version of part (c): the displayed differential expression is set equal to zero. This supplies a complete equation for classification.

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3. Original problem

   Determine if the given function is a solution to the given differential equations.

   1. 
   2. 
   3. 
   4. 

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4. Original problem

   Use separation of variables to solve the following differential equations.

   1. 
   2. 
   3. 

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5. Original problem

   Compute the given partial derivatives.

   1. 
   2. 
   3. 
   4. 

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6. Added problem

   Find $\mathrm{dy}/\mathrm{dx}$ implicitly for $x^2+y^2=25$. State where your formula applies.

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7. Added problem

   Find $\mathrm{dy}/\mathrm{dx}$ implicitly for $\mathrm{xy}=1$.

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8. Added problem

   Implicitly differentiate $x^3+y^3=6\mathrm{xy}$.

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9. Added problem

   Find $\mathrm{dy}/\mathrm{dx}$ implicitly for $\ln \left(\mathrm{xy}\right)=x$.

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10. Added problem

    Find $\mathrm{dy}/\mathrm{dx}$ implicitly for $\sin y=x^2$.

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11. Added problem

    State the order and decide whether $y″+\mathrm{xy}\prime +y=\sin x$ is linear.

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12. Added problem

    State the order and decide whether $(y\prime {)}^2+y=0$ is linear.

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13. Added problem

    State the order and decide whether $y‴+y^2=0$ is linear.

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14. Added problem

    Is $y={\mathrm{Ce}}^{-2x}$ a solution of $y\prime +2y=0$? Verify by substitution.

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15. Added problem

    Solve $y″=6x$ by direct integration. Include all integration constants.

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16. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=3y$ by separation of variables.

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17. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=x/y$ and state the restrictions on the solution.

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18. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=2\mathrm{xy}/(1+x^2)$.

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19. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=y(1-y)$. Include equilibrium solutions and domain restrictions.

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20. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=y^2$ and include all equilibrium solutions.

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21. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=\mathrm{xy}$ with $y\left(0\right)=2$.

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22. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=(1+y^2)\cos x$. State the interval restriction.

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23. Added problem

    Solve $\mathrm{dy}/\mathrm{dx}=x^3/(1+y^2)$. An implicit solution is acceptable.

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24. Added problem

    For $f(x,y)=x^{2}y+3y^2$, find both first partial derivatives.

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25. Added problem

    Find both first partial derivatives of $f(x,y)=\sin \left(\mathrm{xy}\right)$.

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26. Added problem

    Compute both mixed second partial derivatives of $f(x,y)=x^2{y}^3$.

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27. Added problem

    Find the first partial derivatives of $f(x,y)=x/(x+y)$.

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28. Added problem

    For $f(x,y)=e^{\mathrm{xy}}$, find $f_{\mathrm{xx}}$ and $f_{\mathrm{xy}}$.

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29. Added problem

    Find the first partial derivatives and domain of $f(x,y)=\ln (x^2+y^2)$.

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30. Added problem

    For $f(x,y,z)=x^2{\mathrm{yz}}^3$, compute the mixed derivative taking one derivative in each of z, y, and x.

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