4.5E: Applications to Curves (Exercises)

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Jun 10, 2024
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- 4.5: Applications to Curves
- 5: Linear Second Order Equations

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Q4.5.1

In Exercises 4.5.1-4.5.8 find a first order differential equation for the given family of curves.

1. $y(x^2+y^2)=c$

2. $e^{xy}=cy$

3. $\ln |xy|=c(x^2+y^2)$

$y=x^{1/2}+cx$

$y=e^{x^2}+ce^{-x^2}$

${y=x^3+{c\over x}}$

$y=\sin x+ce^x$

$y=e^x+c(1+x^2)$

Q4.5.2

9. Show that the family of circles

$(x-x_0)^2+y^2=1,\;-\infty<x_0<\infty,\nonumber$ can be obtained by joining integral curves of two first order differential equations. More specifically, find differential equations for the families of semicircles

$(x-x_0)^2+y^2=1,\; x_0<x<x_0+1,\;-\infty<x_0<\infty,\nonumber$

$(x-x_0)^2+y^2=1,\; x_0-1<x<x_0,\;-\infty<x_0<\infty.\nonumber$

10. Suppose $f$ and $g$ are differentiable for all $x$ . Find a differential equation for the family of functions $y=f+cg$ ( $c$ =constant).

Q4.5.3

In Exercises 4.5.11-4.5.13 find a first order differential equation for the given family of curves.

11. Lines through a given point $(x_0,y_0)$ .

12. Circles through $(-1,0)$ and $(1,0)$ .

13. Circles through $(0,0)$ and $(0,2)$ .

Q4.5.4

14. Use the method Example 4.5.6 (a) to find the equations of lines through the given points tangent to the parabola $y=x^2$ . Also, find the points of tangency.

$(5,9)$
$(6,11)$
$(-6,20)$
$(-3,5)$

15.

Show that the equation of the line tangent to the circle $x^2+y^2=1 \tag{A}$ at a point $(x_0,y_0)$ on the circle is $y={1-x_0x\over y_0}\quad \text{if} \quad x_0\ne\pm1. \tag{B}$
Show that if $y'$ is the slope of a nonvertical tangent line to the circle (A) and $(x,y)$ is a point on the tangent line then $(y')^2(x^2-1)-2xyy'+y^2-1=0. \tag{C}$
Show that the segment of the tangent line (B) on which $(x-x_0)/y_0>0$ is an integral curve of the differential equation $y'={xy-\sqrt{x^2+y^2-1}\over x^2-1}, \tag{D}$ while the segment on which $(x-x_0)/y_0<0$ is an integral curve of the differential equation $y'={xy+\sqrt{x^2+y^2-1}\over x^2-1}. \tag{E}$ HINT: Use the quadratic formula to solve (C) for $y'$ . Then substitute (B) for $y$ and choose the $\pm$ sign in the quadratic formula so that the resulting expression for $y'$ reduces to the known slope $y'=-x_{0}/y_{0}$
Show that the upper and lower semicircles of (A) are also integral curves of (D) and (E).
Find the equations of two lines through (5,5) tangent to the circle (A), and find the points of tangency.

16.

Show that the equation of the line tangent to the parabola $x=y^2 \tag{A}$ at a point $(x_0,y_0)\ne(0,0)$ on the parabola is $y={y_0\over2}+{x\over2y_0}. \tag{B}$
Show that if $y'$ is the slope of a nonvertical tangent line to the parabola (A) and $(x,y)$ is a point on the tangent line then $4x^2(y')^2-4xyy'+x=0. \tag{C}$
Show that the segment of the tangent line defined in (a) on which $x>x_0$ is an integral curve of the differential equation $y'={y+\sqrt{y^2-x}\over2x}, \tag{D}$ while the segment on which $x<x_0$ is an integral curve of the differential equation $y'={y-\sqrt{y^2-x}\over2x}, \tag{E}$ HINT: Use the quadratic formula to solve (c) for $y'$ . Then substitute (B) for y and choose the $\pm$ sign in the quadratic formula so that the resulting expression for $y'$ reduces to the known slope of $y'=\frac{1}{2y_{0}}$
Show that the upper and lower halves of the parabola (A), given by $y=\sqrt x$ and $y=-\sqrt x$ for $x>0$ , are also integral curves of (D) and (E).

17. Use the results of Exercise 4.5.16 to find the equations of two lines tangent to the parabola $x=y^2$ and passing through the given point. Also find the points of tangency.

$(-5,2)$
$(-4,0)$
$(7,4)$
$(5,-3)$

Find a curve $y=y(x)$ through (1,2) such that the tangent to the curve at any point $(x_0,y(x_0))$ intersects the $x$ axis at ${x_I=x_0/2}$ .

19. Find all curves $y=y(x)$ such that the tangent to the curve at any point $(x_0,y(x_0))$ intersects the $x$ axis at $x_I=x^3_0$ .

20. Find all curves $y=y(x)$ such that the tangent to the curve at any point passes through a given point $(x_1,y_1)$ .

21. Find a curve $y=y(x)$ through $(1,-1)$ such that the tangent to the curve at any point $(x_0,y(x_0))$ intersects the $y$ axis at $y_I=x^3_0$ .

22. Find all curves $y=y(x)$ such that the tangent to the curve at any point $(x_0,y(x_0))$ intersects the $y$ axis at $y_I=x_0$ .

23. Find a curve $y=y(x)$ through $(0,2)$ such that the normal to the curve at any point $(x_0,y(x_0))$ intersects the $x$ axis at $x_I=x_0+1$ .

24. Find a curve $y=y(x)$ through $(2,1)$ such that the normal to the curve at any point $(x_0,y(x_0))$ intersects the $y$ axis at $y_I=2y(x_0)$ .

Q4.5.5

In Exercises 4.5.25-2.5.29 find the orthogonal trajectories of the given family of curves.

25.

$x^2+2y^2=c^2$

26. $x^2+4xy+y^2=c$

27. $y=ce^{2x}$

28. $xye^{x^2}=c$

29. ${y={ce^x\over x}}$

Q4.5.6

30. Find a curve through $(-1,3)$ orthogonal to every parabola of the form

$y=1+cx^2\nonumber$ that it intersects. Which of these parabolas does the desired curve intersect?

31. Show that the orthogonal trajectories of

$x^2+2axy+y^2=c\nonumber$ satisfy

$|y-x|^{a+1}|y+x|^{a-1}=k.\nonumber$

32. If lines $L$ and $L_1$ intersect at $(x_0,y_0)$ and $\alpha$ is the smallest angle through which $L$ must be rotated counterclockwise about $(x_0,y_0)$ to bring it into coincidence with $L_1$ , we say that $\alpha$ is the angle from $L$ to $L_1$ ; thus, $0\le\alpha<\pi$ . If $L$ and $L_1$ are tangents to curves $C$ and $C_1$ , respectively, that intersect at $(x_0,y_0)$ , we say that $C_1$ intersects $C$ at the angle $\alpha$ . Use the identity

$\tan(A+B)={\tan A+\tan B\over1-\tan A\tan B}\nonumber$ to show that if

$C$ and

$C_1$ are intersecting integral curves of

$y'=f(x,y) \quad \text{and} \quad y'={f(x,y)+\tan\alpha\over 1-f(x,y)\tan\alpha} \quad\left( \alpha \ne {\pi\over2}\right),\nonumber$ respectively, then

$C_1$ intersects

$C$ at the angle

$\alpha$ .

33. Use the result of Exercise 4.5.32 to find a family of curves that intersect every nonvertical line through the origin at the angle $\alpha=\pi/4$ .

34. Use the result of Exercise 4.5.32 to find a family of curves that intersect every circle centered at the origin at a given angle $\alpha \ne \pi/2$ .

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