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The general solution of the differential equation \(\dfrac{dy}{dx}=xy\) is __________.

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True or False If \(f(tx,ty)=t^{k}f(x,y),\) for all \(t>0\), where \(k\) is a real number, then the function \(f\) is said to be homogeneous of degree \(k\) in \(x\) and \(y.\)

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If \(F(x,y,C)=0\) and \(G(x,y,K)=0\) are one-parameter families of curves in which each member of one family intersects the members of the other family orthogonally, then the two families are said to be orthogonal __________ of each other.

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If \(M(x,y)\,dx+N(x,y)\,dy=0\) is a homogeneous differential equation, then it can be transformed into an equation whose variables are separable by using the substitution _________ , where _________ is a differentiable function of \(x\).

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\(\dfrac{dy}{dx}=x\sec y\)

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\(\cos y\dfrac{dy}{dx}=\dfrac{2}{x}\)

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\(\dfrac{dy}{dx}=e^{y-x}\)

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\(x\dfrac{dy}{dx}+2y=5\)

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\(\dfrac{dy}{dx}=\cos x\dfrac{dy}{dx}-y\sin x\)

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\(\dfrac{dy}{dx}=x+y \dfrac{dy}{dx}\)

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\(\dfrac{dy}{dx}+xy=x\)

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\((3x+1)\,dx+e^{x+y}dy=0\)

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\(\ln x\dfrac{dx}{dy}=\dfrac{x}{y}\)

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\(\dfrac{dy}{dx}=\dfrac{x+2}{2-y}\)

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\(\dfrac{dy}{dx}=x\sec y,\quad y=\dfrac{\pi }{4}\) when \(x=1\)

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\(\cos y\dfrac{dy}{dx}=\dfrac{2}{x},\quad y=\dfrac{\pi }{3}\), when \(x=-1\)

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\(\dfrac{dy}{dx}=e^{y-x};\quad y=0\) when \(x=0\)

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\(x\dfrac{dy}{dx}+2y=5;\quad y=1\) when \(x=-1\)

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\(\dfrac{dy}{dx}=\cos x\dfrac{dy}{dx}-y\sin x;\quad y=\dfrac{1}{2}\) when \(x=\pi \)

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\(\dfrac{dy}{dx}=x+y \dfrac{dy}{dx},\quad y=1\) when \(x=2\)

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\(f(x,y)=2x^{2}-3xy-y^{2}\)

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\(f(x,y)=x^{3}-xy^{2}+y^{3}\)

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\(f(x,y)=x^{3}-xy+y^{3}\)

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\(f(x,y)=x^{2}-xy^{2}+y^{2}\)

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\(f(x,y)=2x+\sqrt{x^{2}+y^{2}}\)

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\(f(x,y)=\sqrt{x+y}\)

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\(f(x,y)=\tan \left( \dfrac{3x}{y}\right)\)

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\(f(x,y)=e^{x/y}\)

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\(f(x,y)=\ln \dfrac{x}{y}\)

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\(f(x,y)=x\ln x-x\ln y\)

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\((x-y)\,dx+xdy=0\)

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\((x+y)\,dx+x\,dy=0\)

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\((x^{2}\,{+}\,y^{2})\,dx\,{+}\,(x^{2}\,{-}\,xy)\,dy\,{=}\,0\)

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\(xy\,dx\,{+}\,(x^{2}\,{+}\,y^{2})\,dy=0\)

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\(\dfrac{dy}{dx}=\dfrac{y-x}{y+x}\)

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\(\dfrac{dy}{dx}=\dfrac{x+2y}{2x+y}\)

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\(x\dfrac{dy}{dx} = x+y\)

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\(x(x^2\,{-}\,y^2) \dfrac{dy}{dx}\,{-}\,y (x^2\,{+}\,y^2) =0\)

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\(\dfrac{dy}{dx}=\dfrac{2xy}{x^{2}+y^{2}}\)

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\(\dfrac{dy}{dx}=\dfrac{x+2y}{2x-y}\)

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\(x^{2}\dfrac{dy}{dx}=x^{2}+xy-4y^{2};\quad \ y=1\) when \(x=-1\)

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\(x(x^{2}-y^{2})\dfrac{dy}{dx}-y(x^{2}+y^{2})=0;\quad y=-2\) when \(x=-1\)

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\(xy=C\)

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\(y^{2}=2(C-x)\)

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\(y=Cx^{2}\)

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\(x^{2} = Cy^3\)

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\(\dfrac{dy}{dx}=\dfrac{y-x-3}{y+x+4}\)

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\(\dfrac{dy}{dx}=\dfrac{x+2y-3}{2x-y+1};\quad y=4\) when \(x=3\)

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Orthogonal Families Find the orthogonal trajectories of the family of parabolas \(y^{2}=4Cx\).

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Orthogonal Trajectories Find the orthogonal trajectories of all circles tangent to the \(x\)-axis at \((3,0)\).

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Newton's Law of Heating  Newton's Law of Heating states that the rate of increase of the temperature \(u\) of an object immersed in a medium of constant temperature \(T\) is proportional to the difference \(T-u\). The differential equation that models the temperature of the object at time \(t\) is \[ \begin{equation*} \frac{du}{dt}=k(T-u) \end{equation*} \]

where \(k\) is a positive constant.

1. Find a solution of this differential equation.
2. Find the particular solution that satisfies the initial condition \(u(0)=u_{0}\).
3. A metal bar with initial temperature \(25{{}^{\circ}{\rm C}}\) is dropped into a container of boiling water \((100{{}^{\circ}{\rm C}})\). After \(5\) seconds, the temperature of the bar is \(35{{}^{\circ}{\rm C}}\). What will the temperature of the bar be after \(1\) min?
4. Graph the particular solution to the differential equation found in (c).
5. Use the graph to approximate how long will it take for the temperature of the bar to be within \(0.5{{}^{\circ}{\rm C}}\) of the temperature of the boiling water.

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Newton's Law of Cooling According to Newton's Law of Cooling, the rate at which a body cools is proportional to the difference between the temperature of the body and the temperature of the surrounding medium. A certain steel bar has a temperature of \(1230{{}^{\circ}{\rm C}}\) and cools to \(1030{{}^{\circ}{\rm C}}\) in \(10\) min when the surrounding temperature is \(30{{}^{\circ}{\rm C}}\).

1. How long will it take for the bar to cool to \(80{{}^{\circ}{\rm C}}\)?
2. Graph the temperature of the steel bar over the interval \(t=0\) to \(t=240\) (i.e., for the first 4 h).
3. Use the graph to approximate when the steel bar cools to \(50{{}^{\circ}{\rm C}}\).

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Sales Growth The annual sales of a new company are expected to grow at a rate that is proportional to the difference between the sales and an upper limit of $\(20\) million. Sales are $\(0\) initially and are $\(3\) million for the second year.

1. Determine the annual sales at any time \(t\).
2. According to the model, what will sales be during the eighth year of operations?
3. How long will it take for sales to reach $\(12\) million?
4. Graph the annual sales of the company for its first \(20\) years of operation.

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Bacteria Growth A culture of bacteria is growing in a medium that can support a maximum of \(1,000,000\) bacteria. The rate of growth of the population at time \(t\) is proportional to the difference between \(1,000,000\) and the number present at time \(t\). The culture contains \(1000\) bacteria initially, and after 1 h there are \(1500\) bacteria.

1. How many bacteria will there be after \(5\) h?
2. When will the culture contain \(9950\) bacteria?

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Learning Curve The number of words per minute \(W\) that a person can text increases with practice. Assume that the rate of change of \(W\) is proportional to the difference between an upper limit of 150 words per minute and \(W\). That is, the rate of change is proportional to \(150-W\). Suppose a beginner cannot text at all, that is, \(W=0\) when \(t=0,\) and can text \(30\) words per minute after \(10\) h of practice.

1. How many words per minute can he text after \(25\) h of practice?
2. How many hours of practice will be required in order for him to text \(100\) words per minute?
3. Graph the number of words per minute \(W\) a beginner can text as a function of hours of practice.

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Drug Concentration A drug is injected into a patient's bloodstream at a constant rate of \(r\) mg/s. Simultaneously, the drug is removed from the bloodstream at a rate proportional to the amount \(y(t)\) present at time \(t\).

1. Find a differential equation that models this problem.
2. Find a solution of the differential equation in (a).
3. Find the particular solution that satisfies the initial condition \(y(0)=0\).

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Friction The frictional force \(f\) on an object sliding over a surface depends on the speed \(v\) of the object and can be modeled by the function \(f=-Av\), where \(A\) is a positive constant that depends on the two surfaces. Note:The force is negative because the direction of the friction is opposite to the direction of the velocity.

1. Express the frictional force \(f\) as a differential equation by using Newton's Second Law, \(F=ma\).
2. Solve the differential equation found in (a) for \(v(t)\). Assume the object slides over the surface with an initial speed \(v_{0}\).
3. Suppose that the frictional force on a 25-kg sled is \(150\)N when the speed of the sled is \(10\) m/s. Determine the value of the constant \(A\).
4. How long will it take the sled to slow down to \(10\)% of its initial speed?
5. How far will the sled travel while slowing down to \(1.0\) m/s ? Assume an initial speed of \(10\) m/s.

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Friction The frictional force \(f\) on an object sliding over a surface depends on the speed \(v\) of the object and can be modeled by the function \(f=-B\sqrt{v}\), where \(B\) is a positive constant that depends on the two surfaces. Note: The force is negative because the direction of the friction is opposite to the direction of the velocity.

1. Express the frictional force \(f\) as a differential equation by using Newton's Second Law, \(F=ma\).
2. Solve the differential equation found in (a) for \(v(t)\). Assume the object slides over the surface with an initial speed \(v_{0}\).
3. Suppose that the frictional force on a 25-kg sled is \(150\)N when the speed of the sled is \(10\) m/s. Determine the value of the constant \(B\).
4. How long will it take the sled to slow down to \(10\)% of its initial speed?
5. How far will the sled travel while slowing down to \(1.0\) m/s? Assume an initial speed of \(10\) m/s.

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Electronics: Discharging a Capacitor In a circuit, a capacitor \(C\) carries an initial charge \(q_{0}\) and discharges through a resistor \(R.\) Using Kirchhoff's rules with this circuit, we obtain the differential equation \[ \begin{equation*} \dfrac{dq}{dt}+\dfrac{q}{RC}=0 \end{equation*} \]

where \(q\) is the charge, in Coulombs, on the capacitor at any time \(t\), and \(R\) and \(C\) are constants.

1. Solve the differential equation for the charge \(q\) as a function of time \(t\).
2. In a typical laboratory circuit, \(R=150\) Ω (ohms) and \(C=5.0\) µF (microfarads). How long will it take the discharging capacitor to lose \(99\)% of its initial charge?
   [Hint: The product (1Ω) (1F) = 1 second.]

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Air Resistance At high speeds, air resistance (drag) can be modeled by \(F_{\rm drag}=-Av^{2}\), where \(A\) is a positive constant that depends on the shape of the object, its surface texture, and the conditions of the air. Note: The force is negative because the direction of the drag is opposite to the direction of the velocity. Suppose, in a test facility, a projectile of mass \(m\) is launched with initial horizontal speed \(v_{0}\) on a frictionless horizontal track. The only force opposing the motion of the projectile is the drag.

1. Express the drag \(F_{\rm drag}\) as a differential equation by using Newton's Second Law, \(F=ma\).
2. Solve the differential equation found in (a) for \(v(t)\).
3. What happens to the speed of the projectile as \(t\) becomes arbitrarily large?

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Orthogonal Families Find the orthogonal trajectories of the family of hyperbolas \(\dfrac{x^{2}}{C^{2}}-\dfrac{y^{2}}{4-C^{2}}=1\), where \(0 < C < 2\).

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Orthogonal Trajectories Find the orthogonal trajectories of the family of parabolas with a common vertex at \((1,2)\) having a vertical axis. Graph the member of each family that contains the point \((2, 3)\).

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Velocity of a Projectile Near Earth's surface the attraction due to gravity is practically constant, but according to Newton's law, the force of attraction exerted by Earth on a given body is \(\dfrac{k}{x^{2}}\), where \(x\) is the distance of the body from the center of the Earth.

1. Show that if a projectile is shot upward from Earth's surface with initial velocity \(v_{0}\), and if air resistance is neglected, then \[ \begin{equation*} v^{2}=v_{0}^{2}-2Rg+\dfrac{2R^{2}g}{x} \end{equation*} \] where \(R=3960\) mi is the radius of Earth and \(g\) is the acceleration due to gravity. Hint: \(\dfrac{d^{2}x}{dt^{2}}= \dfrac{dv}{dt}=\dfrac{dv}{dx}\dfrac{dx}{dt}=v\dfrac{dv}{dx}\)
2. Find the smallest value of \(v_{0}\) necessary to keep the projectile going indefinitely (neglecting the attractive forces due to celestial bodies other than Earth). This is called the escape velocity for Earth.

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Rate of Flow From physics, it is known that the rate of flow of water from a tank through an orifice of area \(A\) square feet is approximately \(\dfrac{dV}{dt}=-kA\sqrt{2gh}\), where \(k=0.6\), \(g=32\) ft/s\(^2,\) and \(h\) is the height of the water level, in feet, above the orifice. Suppose a cylindrical tank with height \(6\) ft and diameter \(3\) ft is filled with water. At time \(t=0\), a valve in the bottom of the tank is opened and the water begins draining through a circular hole of diameter \(1\) in.

1. Express the height of the water in the tank as a function of time \(t\).
2. How much time is required to empty the tank?

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Rate of Flow If the water in a tank leaks out through a small hole in the bottom, then the rate of flow is proportional to the square root of the height of the water in the tank. Prove that if the tank is a cylinder with a vertical axis, then the time required for three-fourths of the water to leak out is equal to the time required for the remaining one-fourth of the water to leak out.

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Water Level A cylindrical tank has a leak in the bottom, and water flows out at a rate proportional to the pressure at the bottom. If the tank loses \(2\)% of its water in \(24\) h, when will it be half empty?