Kenneth Kuttler

572 CHAPTER 29. FIRST ORDER SCALAR ODE

(a) x2dx+(y2 +1

)dy = 0, (1,1)

(b) xydx+(y2 +1

)dy = 0, (1,1)

)dy = 0, (1,−1)

(d) ydx+(y2 +1

)xdy = 0, (1,2)

(e) 0 = cos(y)dx+ tan(x)dy,(

2 ,π

)(f) xydx =

(y2 +1

)dy, (1,1)

(g) xydx =(y2−1

)dy, (2,1)

22. Find all integral curves of the equation yxdx+ e−x2dy = 0. Graph several.

23. Find all integral curves of the equation yxdx+ 1ln(1+x2)

y3dy = 0. Graph several.

24. Give the integral curves to the equation v′ = g− kv2 mentioned above where g isacceleration of gravity and k a positive constant.

25. You have a collection of hyperbolas x2− y2 = C where each choice of C leads to adifferent hyperbola. Find another collection of curves which intersect these at a rightangle. Hint: Say you have f (x,y) =C is one of these. If you are at a point where therelation defines y as a function of x, and (x,y) is a point on one of these hyperbolasjust mentioned, then dy

dx should have a relation to the tangent line to x2− y2 = C.Since the two curves are to be perpendicular, you should have the product of theirslopes equal to −1. Thus

(dydx

)(xy

)=−1.

26. Generalize the above problem. Suppose you have a family of level curves f (x,y) =Cand you want another family of curves which is perpendicular to this family of curvesat every point of intersection. Find a differential equation which will express thiscondition. Recall that two curves are perpendicular if the products of the slopes ofthe tangent lines to the two curves equals −1.

27. Find and determine the stability of the equilibrium points for the following separableequations.

(a) y′ = y2 (y−1)(b) y′ = (y+1)(y−1)(y+2)(c) y′ = sin(y)(d) y′ = cos(y)

(e) y′ = ln(1+ y2

)(f) y′ = e2y−1

(g) y′ = 1− ey2

28. The force on an object of mass m acted on by the earth having mass M is givenby Newton’s formula kmM/r2 where k is the gravitation constant first calculated byCavendish4 in 1798. Letting R be the radius of the earth and letting g denote theacceleration of gravity on the earth’s surface, show that kM = R2g. Now suppose alarge gun having its muzzle at the surface of the earth is fired away from the centerof the earth such that the projectile has velocity v0. Explain why

dvdt

=− R2g

(R+ r)2

4For about 100 years, since the time Newton claimed the existence of this gravitation constant, no one knewwhat it was. Henry Cavendish did an extremely sensitive experiment in 1797-1798 to determine it. It involvedlead balls mirrors telescopes and a torsion balance. He was a chemist who also found ways to make hydrogen. Hedid many other very precise experiments in physics and chemistry.

57222.23.24.25.26.27.28.CHAPTER 29. FIRST ORDER SCALAR ODE(a) x°dx+(y?+1)dy=(b) xydx+ (y?+1)dy=(c) xydx+ (y?+1)dy=(d) ydx+ (y? +1) xdy, (ec) 0 =cos(y)dx+ tan (x) dy, (%, 7)(f) xydx = (y+ 1) dy, (1,1),2) (g) xydx = (y?— 1) dy, (2,1)9Find all integral curves of the equation yxdx + ew dy = 0. Graph several.Find all integral curves of the equation yxdx + 5 mn) ®) y>dy = 0. Graph several.Give the integral curves to the equation v’ = g—kv* mentioned above where g isacceleration of gravity and k a positive constant.You have a collection of hyperbolas x” — y? = C where each choice of C leads to adifferent hyperbola. Find another collection of curves which intersect these at a rightangle. Hint: Say you have f (x,y) =C is one of these. If you are at a point where therelation defines y as a function of x, and (x,y) is a point on one of these hyperbolasjust mentioned, then 2 should have a relation to the tangent line to x7 — y? =C.Since the two curves are to be perpendicular, you should have the product of theirslopes equal to —1. Thus (4) («) =-l.Generalize the above problem. Suppose you have a family of level curves f (x,y) =Cand you want another family of curves which is perpendicular to this family of curvesat every point of intersection. Find a differential equation which will express thiscondition. Recall that two curves are perpendicular if the products of the slopes ofthe tangent lines to the two curves equals —1.Find and determine the stability of the equilibrium points for the following separableequations.(a) y =y"(y-1) (e) y’ =In(1+y”’)(b) Y= VtYO-DO+2) () y=e"-1(c) y’ =sin(y) ;(d) y’ =cos(y) (g) y=1-eThe force on an object of mass m acted on by the earth having mass M is givenby Newton’s formula kmM/r? where k is the gravitation constant first calculated byCavendish* in 1798. Letting R be the radius of the earth and letting g denote theacceleration of gravity on the earth’s surface, show that kM = R7g. Now suppose alarge gun having its muzzle at the surface of the earth is fired away from the centerof the earth such that the projectile has velocity vg. Explain whydy __Redt = (R+r)°4For about 100 years, since the time Newton claimed the existence of this gravitation constant, no one knewwhat it was. Henry Cavendish did an extremely sensitive experiment in 1797-1798 to determine it. It involvedlead balls mirrors telescopes and a torsion balance. He was a chemist who also found ways to make hydrogen. Hedid many other very precise experiments in physics and chemistry.