Solved: In 1 and 2, fill in the blank and then write this result as a linear first-order

In Problems 1 and 2, fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol \(c_{1}\) and has the form dy/dx = f (x, y). The symbols \(c_{1}\) and k represent constants. \(\frac{d}{d x} c_{1} e^{k x}\) = _____________________________

In Problems 1 and 2, fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol \(c_{1}\) and has the form dy/dx = f (x, y). The symbols \(c_{1}\) and k represent constants. \(\frac{d}{d x}\left(5+c_{1} e^{-2 x}\right)\) = _____________________________

In Problems 3 and 4, fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F( y, y) 0. The symbols c1, c2, and k represent constants. \(\frac{d^{2}}{d x^{2}}\left(c_{1} \cos k x+c_{2} \sin k x\right)\) = _____________________________

In Problems 3 and 4, fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F( y, y) 0. The symbols c1, c2, and k represent constants. \(\frac{d^{2}}{d x^{2}}\left(c_{1} \cosh k x+c_{2} \sinh k x\right)\) = _____________________________

In Problems 5 and 6, compute \(y^{\prime} \text { and } y^{\prime \prime}\) and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols \(c_{1}\) and \(c_{2}\) and has the form \(F\left(y, y^{\prime}, y^{\prime \prime}\right)=0\). The symbols \(c_{1}\) and \(c_{2}\) represent constants. \(y=c_{1} e^{x}+c_{2} x e^{x}\)

In Problems 5 and 6, compute \(y^{\prime} \text { and } y^{\prime \prime}\) and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols \(c_{1}\) and \(c_{2}\) and has the form \(F\left(y, y^{\prime}, y^{\prime \prime}\right)=0\). The symbols \(c_{1}\) and \(c_{2}\) represent constants. \(y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x\)

In Problems 7–12, match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 \(x y^{\prime}=2 y\)

In Problems 7–12, match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 \(y^{\prime}=2\)

In Problems 7–12, match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 \(y^{\prime}=2 y-4\)

In Problems 7–12, match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 \(x y^{\prime}=y\)

In Problems 7–12, match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 \(y^{\prime \prime}+9 y=18\)

In Problems 7–12, match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 \(x y^{\prime \prime}-y^{\prime}=0\)

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation. \(y^{\prime \prime}=y^{\prime}\)

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation. \(y^{\prime}=y(y-3)\)

In Problems 15 and 16, interpret each statement as a differential equation. On the graph of \(y=\phi(x)\), the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin.

In Problems 15 and 16, interpret each statement as a differential equation. On the graph of \(y=\phi(x)\), the rate at which the slope changes with respect to x at a point P(x, y) is the negative of the slope of the tangent line at P(x, y).

(a) Give the domain of the function \(y=x^{2 / 3}\). (b) Give the largest interval I of definition over which \(y=x^{2 / 3}\) is a solution of the differential equation \(3 x y^{\prime}-2 y=0\).

(a) Verify that the one-parameter family \(y^{2}-2 y=x^{2}-x+c\) is an implicit solution of the differential equation \((2 y-2) y^{\prime}=2 x-1\). (b) Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0) = 1. (c) Use your result in part (b) to find an explicit function \(y=\phi(x)\) that satisfies y(0) = 1. Give the domain of \(\phi\). Is \(y=\phi(x)\) a solution of the initial-value problem? If so, give its interval I of definition; if not, explain.

Given that \(y=-\frac{2}{x}+x\) is a solution of the DE \(x y^{\prime}+y=2 x\). Find \(x_{0}\) and the largest interval I for which y(x) is a solution of the IVP \(x y^{\prime}+y=2 x\), \(y\left(x_{0}\right)=1\).

Suppose that y(x) denotes a solution of the initial-value problem \(y^{\prime}=x^{2}+y^{2}\), y(1) = -1 and that y(x) possesses at least a second derivative at x = 1. In some neighborhood of x 1, use the DE to determine whether y(x) is increasing or decreasing, and whether the graph y(x) is concave up or concave down.

A differential equation may possess more than one family of solutions. (a) Plot different members of the families \(y=\phi_{1}(x)=x^{2}+c_{1}\) and \(y=\phi_{2}(x)=-x^{2}+c_{2}\). (b) Verify that \(y=\phi_{1}(x)\) and \(y=\phi_{2}(x)\) are two solutions of the nonlinear first-order differential equation \(\left(y^{\prime}\right)^{2}=4 x^{2}\). (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).

What is the slope of the tangent line to the graph of the solution of \(y^{\prime}=6 \sqrt{y}+5 x^{3}\) that passes through (-1, 4)?

In Problems 23–26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. \(y^{\prime \prime}+y=2 \cos x-2 \sin x\); y = x sin x + x cos x

In Problems 23–26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. \(y^{\prime \prime}+y \sec x\); y = x sin x + (cos x) ln(cos x)

In Problems 23–26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. \(x^{2} y^{\prime \prime}+x y^{\prime}+y=0\); y = sin(ln x)

In Problems 23–26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. \(x^{2} y^{\prime \prime}+x y^{\prime}+y=\sec (\ln x)\); y = cos(ln x) ln(cos(ln x)) + (ln x) sin(ln x)

In Problems 27–30, use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. \(\frac{d y}{d x}+(\sin x) y=x\); \(y=e^{\cos x} \int_{0}^{x} t e^{-\cos t} d t\)

In Problems 27–30, use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. \(\frac{d y}{d x}-2 x y=e^{x}\); \(y=e^{x^{2}} \int_{0}^{x} e^{t-t^{2}} d t\)

In Problems 27–30, use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. \(x^{2} y^{\prime \prime}+\left(x^{2}-x\right) y^{\prime}+(1-x) y=0); \(y=x \int_{1}^{x} \frac{e^{-t}}{t} d t\)

In Problems 27–30, use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. \(y^{\prime \prime}+y=e^{x^{2}}\); \(y=\sin x \int_{0}^{x} e^{t^{2}} \cos t d t-\cos x \int_{0}^{x} e^{t^{2}} \sin t d t\)

In Problems 31–34, verify that the indicated expression is an implicit solution of the given differential equation. \(x \frac{d y}{d x}+y=\frac{1}{y^{2}}\); \(x^{3} y^{3}=x^{3}+5\)

In Problems 31–34, verify that the indicated expression is an implicit solution of the given differential equation. \(\left(\frac{d y}{d x}\right)^{2}+1=\frac{1}{y^{2}}\); \((x-7)^{2}+y^{2}=1\)

In Problems 31–34, verify that the indicated expression is an implicit solution of the given differential equation. \(y^{\prime \prime}=2 y\left(y^{\prime}\right)^{3}\); \(y^{3}+3 y=2-3 x\)

In Problems 31–34, verify that the indicated expression is an implicit solution of the given differential equation. \((1+x y) y^{\prime}+y^{2}=0\); \(y=e^{-x y}\)

In Problems 35–38, \(y=c_{1} e^{-3 x}+c_{2} e^{x}+4 x\) is a two-parameter family of the second-order differential equation \(y^{\prime \prime}+2 y^{\prime}-3 y=-12 x+8\). Find a solution of the second-order initial-value problem consisting of this differential equation and the given initial conditions. y(0) = 0, \(y^{\prime}(0)=0\)

In Problems 35–38, \(y=c_{1} e^{-3 x}+c_{2} e^{x}+4 x\) is a two-parameter family of the second-order differential equation \(y^{\prime \prime}+2 y^{\prime}-3 y=-12 x+8\). Find a solution of the second-order initial-value problem consisting of this differential equation and the given initial conditions. y(0) = 5, \(y^{\prime}(0)=-11\)

In Problems 35–38, \(y=c_{1} e^{-3 x}+c_{2} e^{x}+4 x\) is a two-parameter family of the second-order differential equation \(y^{\prime \prime}+2 y^{\prime}-3 y=-12 x+8\). Find a solution of the second-order initial-value problem consisting of this differential equation and the given initial conditions. y(1) = -2, \(y^{\prime}(1)=4\)

In Problems 35–38, \(y=c_{1} e^{-3 x}+c_{2} e^{x}+4 x\) is a two-parameter family of the second-order differential equation \(y^{\prime \prime}+2 y^{\prime}-3 y=-12 x+8\). Find a solution of the second-order initial-value problem consisting of this differential equation and the given initial conditions. y(-1) = 1, \(y^{\prime}(-1)=1\)

In Problem 39 and 40, verify that the function defined by the definite integral is a particular solution of the given differential equation. In both problems, use Leibniz’s rule for the derivative of an integral: \(\frac{d}{d x} \int_{u(x)}^{v(x)} F(x, t) d t=F(x, v(x)) \frac{d v}{d x}-F(x, u(x)) \frac{d u}{d x}+\int_{u(x)}^{v(x)} \frac{\partial}{\partial x} F(x, t) d t\). \(y^{\prime \prime}+9 y=f(x)\); \(y(x)=\frac{1}{3} \int_{0}^{x} f(t) \sin 3(x-t) d t\)

In Problem 39 and 40, verify that the function defined by the definite integral is a particular solution of the given differential equation. In both problems, use Leibniz’s rule for the derivative of an integral: \(\frac{d}{d x} \int_{u(x)}^{v(x)} F(x, t) d t=F(x, v(x)) \frac{d v}{d x}-F(x, u(x)) \frac{d u}{d x}+\int_{u(x)}^{v(x)} \frac{\partial}{\partial x} F(x, t) d t\). \(x y^{\prime \prime}+y^{\prime}-x y=0\); \(y=\int_{0}^{\pi} e^{x \cos t} d t\) [Hint: After computing \(y^{\prime}\) use integration by parts with respect to t.]

The graph of a solution of a second-order initial-value problem \(d^{2} y / d x^{2}=f\left(x, y, y^{\prime}\right)\), \(y(2)=y_{0}\), \(y^{\prime}(2)=y_{1}\), is given in FIGURE 1.R.1. Use the graph to estimate the values of \(y_{0} \text { and } y_{1}\).

A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water, and water leaks from a circular hole of radius 1/2 inch at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole.

A uniform 10-foot-long heavy rope is coiled loosely on the ground. As shown in FIGURE 1.R.2 one end of the rope is pulled vertically upward by means of a constant force of 5 lb. The rope weighs 1 lb/ft. Use Newton’s second law in the form given in (17) in Exercises 1.3 to determine a differential equation for the height x(t) of the end above ground level at time t. Assume that the positive direction is upward.