Exact Differential Equation In Exercises 57 and 58, find all values of such that the

Testing for Exactness In Exercises 1-4, determine whether the differential equation is exact. Explain your reasoning. \(\left(2 x+x y^{2}\right) d x+\left(3+x^{2} y\right) d y=0\) Text Transcription: (2x+xy^2)dx + (3+x^2y) dy=0

Testing for Exactness In Exercises 1-4, determine whether the differential equation is exact. Explain your reasoning. \((1-x y) d x+(y-x y) d y=0\) Text Transcription: (1-x y) d x+(y-x y) d y=0

Testing for Exactness In Exercises 1-4, determine whether the differential equation is exact. Explain your reasoning. \(x \sin y \ d x+x \cos y \ d y=0\) Text Transcription: x sin y dx+x cos y dy=0

Testing for Exactness In Exercises 1-4, determine whether the differential equation is exact. Explain your reasoning. \(y e^{x y} d x+x e^{x y} d y=0\) Text Transcription: y e^xy dx+x e^xy dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \((2 x-3 y) d x+(2 y-3 x) d y=0\) Text Transcription: (2x-3y) dx+(2y-3x) dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(y e^{x} d x+e^{x} d y=0\) Text Transcription: y e^x dx+e^x dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(\left(3 y^{2}+10 x y^{2}\right) d x+\left(6 x y-2+10 x^{2} y\right) d y=0\) Text Transcription: (3x^2+10xy^2)dx+(6xy-2+10x^2y)dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(2 \cos (2 x-y) d x-\cos (2 x-y) d y=0\) Text Transcription: 2 cos (2 x-y) dx-cos (2x-y) dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(\left(4 x^{3}-6 x y^{2}\right) d x+\left(4 y^{3}-6 x y\right) d y=0\) Text Transcription: (4x^3-6xy^2)dx+(4y^3-6xy)dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(2 y^{2} e^{x y^{2}} d x+2 x y e^{x y^{2}} d y=0\) Text Transcription: 2y^2 e^xy^2 dx + 2xye^xy^2 dy=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(\frac{1}{x^{2}+y^{2}}(x d y-y d x)=0\) Text Transcription: 1/x^2+y^2 (x dy-y dx)=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(e^{-\left(x^{2}+y^{2}\right)}(x d x+y d y)=0\) Text Transcription: e^y cos xy [y dx] + (tan xy) dy]=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(\frac{1}{(x-y)^{2}}\left(y^{2} d x+x^{2} d y\right)=0\) Text Transcription: 1/(x-y)^2 (y^2 dx+x^2 dy)=0

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution. \(e^{y} \cos x y[y \ d x+(x+\tan x y) d y]=0\) Text Transcription: e^y cos x y[y d x+(x+tan x y) dy]=0

Graphical and Analytic Analysis In Exercises 15 and 16, (a) sketch an approximate solution of the differential equation satisfying the initial condition on the slope field, (b) find the particular solution that satisfies the initial condition, and (c) use a graphing utility to graph the particular solution. Compare the graph with the sketch in part (a). Differential Equation Initial Condition \((2 x \tan y+5) d x+\left(x^{2} \sec ^{2} y\right) d y=0 \quad\left(\frac{1}{2}, \frac{\pi}{4}\right)\) Text Transcription: (2x tan y+5)dx + (x^2 sec^2 y)dy=0 (1/2, pi/4)

Graphical and Analytic Analysis In Exercises 15 and 16, (a) sketch an approximate solution of the differential equation satisfying the initial condition on the slope field, (b) find the particular solution that satisfies the initial condition, and (c) use a graphing utility to graph the particular solution. Compare the graph with the sketch in part (a). Differential Equation Initial Condition \(\frac{1}{\sqrt{x^{2}+y^{2}}}(x d x+y d y)=0 \quad (4,3)\) Text Transcription: 1/sqrt x^2+y^2 (x dx+y dy)=0

Finding a Particular Solution In Exercises 17-22,find the particular solution that satisfies the initial condition. Differential Equation Initial Condition \(\frac{y}{x-1} d x+[\ln (x-1)+2 y] d y=0\) y(2) = 4 Text Transcription: y/x-1 dx + [ln(x-1) + 2y]dy = 0

Finding a Particular Solution In Exercises 17-22,find the particular solution that satisfies the initial condition. Differential Equation Initial Condition \(\frac{1}{x^{2}+y^{2}}(x d x+y d y)=0\) y(2) = 4 Text Transcription: 1/x^2+y^2 (x dx + y dy) = 0

Finding a Particular Solution In Exercises 17-22,find the particular solution that satisfies the initial condition. Differential Equation Initial Condition \(e^{3 x}(\sin 3 y d x+\cos 3 y d y)=0\) y(2) = \(\pi\) Text Transcription: e^3x(sin 3y dx + cos 3y dy) = 0 pi

Finding a Particular Solution In Exercises 17-22,find the particular solution that satisfies the initial condition. Differential Equation Initial Condition \(\left(x^{2}+y^{2}\right) d x+2 x y \ d y=0\) y(3) = 1 Text Transcription: (x^2+y^2)dx+2xy dy=0

Finding a Particular Solution In Exercises 17-22,find the particular solution that satisfies the initial condition. Differential Equation Initial Condition \(\left(2 x y-9 x^{2}\right) d x+\left(2 y+x^{2}+1\right) d y=0\) y(0) = -3 Text Transcription: (2xy-9x^2)dx+(2y+x^2+1)dy=0

Finding a Particular Solution In Exercises 17-22,find the particular solution that satisfies the initial condition. Differential Equation Initial Condition \(\left(2 x y^{2}+4\right) d x+\left(2 x^{2} y-6\right) d y=0\) y(-1) = 8 Text Transcription: (2xy^2+4)dx+(2x^2y-6)dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(y \ d x-\left(x+6 y^{2}\right) d y=0\) Text Transcription: y dx-(x+6y^2)dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(\left(2 x^{3}+y\right) d x-x \ d y=0\) Text Transcription: (2x^3+y)dx-x dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(\left(5 x^{2}-y\right) d x+x \ d y=0\) Text Transcription: (5x^2-y)dx+x dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(\left(5 x^{2}-y^{2}\right) d x+2 y d y=0\) Text Transcription: (5x^2-y^2)dx+2ydy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \((x+y) d x+\tan x d y=0\) Text Transcription: (x+y) d x+\tan x dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(\left(2 x^{2} y-1\right) d x+x^{3} d y=0\) Text Transcription: (2 x^2 y-1) dx+x^3 dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(y^{2} d x+(x y-1) d y=0\) Text Transcription: y^2 dx+(x y-1) dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(\left(x^{2}+2 x+y\right) d x+2 d y=0\) Text Transcription: (x^2+2x+y)dx+2dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(2 y d x+(x-\sin \sqrt{y}) d y=0\) Text Transcription: 2ydx+(x-sin sqrt y)dy=0

Finding an Integrating Factor In Exercises 23-32,find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. \(\left(-2 y^{3}+1\right) d x+\left(3 x y^{2}+x^{3}\right) d y=0\) Text Transcription: (-2y^3+1)dx+(3xy^2+x^3)dy=0

Using an Integrating Factor In Exercises 33-36, use the integrating factor to find the general solution of the differential equation. Integrating Factor Differential Equation \(u(x, y)=x y^{2}\) \(\left(4 x^{2} y+2 y^{2}\right) d x+\left(3 x^{3}+4 x y\right) d y=0\) Text Transcription: u(x, y)=x y^2 (4x^2y+2y^2)dx+(3x^3+4xy)dy=0

Using an Integrating Factor In Exercises 33-36, use the integrating factor to find the general solution of the differential equation. Integrating Factor Differential Equation \(u(x, y)=x^{2} y\) \(\left(3 y^{2}+5 x^{2} y\right) d x+\left(3 x y+2 x^{3}\right) d y=0\) Text Transcription: u(x, y)=x^2 y (3y^2+5x^2y)dy+(3xy+2x^3)dy=0

Using an Integrating Factor In Exercises 33-36, use the integrating factor to find the general solution of the differential equation. Integrating Factor Differential Equation \(u(x, y)=x^{-2} y^{-3}\) \(\left(-y^{5}+x^{2} y\right) d x+\left(2 x y^{4}-2 x^{3}\right) d y=0\) Text Transcription: u(x, y)=x^-2 y^-3 (-y^5+x^2y)dx+(2xy^4-2x^3)dy=0

Using an Integrating Factor In Exercises 33-36, use the integrating factor to find the general solution of the differential equation. Integrating Factor Differential Equation \(u(x, y)=x^{-2} y^{-2}\) \(-y^{3} d x+\left(x y^{2}-x^{2}\right)d y=0\) Text Transcription: u(x, y)=x^-2 y^-2 -y^3 dx+(xy^2-x^2)dy=0

Integrating Factor Show that each expression is an integrating factor for the differential equation ydx - xdy = 0. (a) \(\frac{1}{x^{2}}\) (b) \(\frac{1}{y^{2}}\) (c) \(\frac{1}{x y}\) (d) \(\frac{1}{x^{2}+y^{2}}\) Text Transcription: 1/x^2 1/y^2 1/xy 1/x^2+y^2

Integrating Factor Show that the differential equation \(\left(a x y^{2}+b y\right) d x+\left(b x^{2} y+a x\right) d y=0\) is exact only when a = b. For \(a \neq b\), show that \(x^{m} y^{n}\) is an integrating factor, where \(m=-\frac{2 b+a}{a+b}, \quad n=-\frac{2 a+b}{a+b}\). Text Transcription: (axy^2+by)dx+(bx^2y+ax)dy=0 a neq b x^m y^n m=-2b+a/a+b, n=-2a+b/a+b

Tangent Curves In Exercises 39-42,use a graphing utility to graph the family of curves tangent to the force field. \(\mathbf{F}(x, y)=\frac{y}{\sqrt{x^{2}+y^{2}}} \mathbf{i}-\frac{x}{\sqrt{x^{2}+y^{2}}} \mathbf{j}\) Text Transcription: F(x,y)=y/sqrt x^2+y^2 i - x/sqrt x^2+y^2 j

Tangent Curves In Exercises 39-42,use a graphing utility to graph the family of curves tangent to the force field. \(\mathbf{F}(x, y)=\frac{x}{\sqrt{x^{2}+y^{2}}} \mathbf{i}-\frac{y}{\sqrt{x^{2}+y^{2}}} \mathbf{j}\) Text Transcription: F(x,y)=x/sqrt x^2+y^2 i - y/x^2+y^2 j

Tangent Curves In Exercises 39-42,use a graphing utility to graph the family of curves tangent to the force field. \(\mathbf{F}(x, y)=4 x^{2} y \mathbf{i}-\left(2 x y^{2}+\frac{x}{y^{2}}\right) \mathbf{j}\) Text Transcription: F(x,y)=4x^2yi-(2xy^2+x/y^2)j

Tangent Curves In Exercises 39-42,use a graphing utility to graph the family of curves tangent to the force field. \(\mathbf{F}(x, y)=\left(1+x^{2}\right) \mathbf{i}-2 x y \mathbf{j}\) Text Transcription: F(x,y)=(1+x^2)i-2xyj

Finding an Equation of a Curve In Exercises 43 and 44, find an equation of the curve with the specified slope passing through the given point. Slope Point \(\frac{d y}{d x}=\frac{y-x}{3 y-x}\) (2, 1) Text Transcription: dy/dx=y-x/3yx

Finding an Equation of a Curve In Exercises 43 and 44, find an equation of the curve with the specified slope passing through the given point. Slope Point \(\frac{d y}{d x}=\frac{-2 x y}{x^{2}+y^{2}}\) (0, 2) Text Transcription: dy/dx=-2xy/x^2+y^2

Cost In a manufacturing process where y = C(x) represents the cost of producing x units, the elasticity of cost is defined as \(E(x)=\frac{\text { marginal cost }}{\text { average cost }}=\frac{C^{\prime}(x)}{C(x) / x}=\frac{x}{y} \frac{d y}{d x}\) Find the cost function when the elasticity function is \(E(x)=\frac{20 x-y}{2 y-10 x}\) where C(100) = 500 and \(x \geq 100\). Text Transcription: E(x)=marginal cost/average cost=C’(x)/C(x)/x=x/y dy/dx E(x)=20x-y/2y-10x x geq 100

HOW DO YOU SEE IT? The graph of the cost function in Exercise 45 is shown below. Use the figure to estimate the limit of the cost function as x approaches 100 from the right.

Euler's Method In Exercises 47 and 48, (a) use Euler's Method and a graphing utility to graph the particular solution of the initial value problem over the indicated interval with the specified value of h and initial condition, (b) find the exact solution of the differential equation analytically, and (c) use a graphing utility to graph the particular solution and compare the result with the graph in part (a). Differential Initial Equation Interval h Condition \(y^{\prime}=\frac{-x y}{x^{2}+y^{2}}\) [2, 4] 0.005 y(2) = 1 Text Transcription: y’=-xy/x^2+y^2

Euler's Method In Exercises 47 and 48, (a) use Euler's Method and a graphing utility to graph the particular solution of the initial value problem over the indicated interval with the specified value of h and initial condition, (b) find the exact solution of the differential equation analytically, and (c) use a graphing utility to graph the particular solution and compare the result with the graph in part (a). Differential Initial Equation Interval h Condition \(y^{\prime}=\frac{6 x+y^{2}}{y(3 y-2 x)}\) [0, 5] 0.2 y(0) = 1 Text Transcription: y’=6x+y^2/y(3y-2x)

Euler's Method Repeat Exercise 47 for h = 1 and discuss how the accuracy of the result changes.

Euler's Method Repeat Exercise 48 for h = 0.5 and discuss how the accuracy of the result changes.

Testing for Exactness Explain how to determine whether a differential equation is exact.

Finding an Integrating Factor Outline the procedure for finding an integrating factor for the differential equation M(x, y) dx + N(x, y) dy = 0.

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The differential equation \(2 x y d x+\left(y^{2}-x^{2}\right) d y=0\) is exact. Text Transcription: 2xy dx + (y^2-x^2)dy=0

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If M dx + Ndy = 0 is exact, then xM dx + xN dy = 0 is also exact.

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If M dx + N dy = 0 is exact, then [f(x) + M] dx + [g(y) + N]dy = 0 is also exact.

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The differential equation f(x) dx + g(y)dy = 0 is exact.

Exact Differential Equation In Exercises 57 and 58, find all values of such that the differential equation is exact. \(\left(x y^{2}+k x^{2} y+x^{3}\right) d x+\left(x^{3}+x^{2} y+y^{2}\right) d y=0\) Text Transcription: (ye^2xy + 2x) dx + (kxe^2xy - 2y) dy = 0

Exact Differential Equation In Exercises 57 and 58, find all values of such that the differential equation is exact. \(\left(y e^{2 x y}+2 x\right) d x+\left(k x e^{2 x y}-2 y\right) d y=0\) Text Transcription: (ye^2xy + 2x) dx + (kxe^2xy - 2y) dy = 0

Exact Differential Equation Find all nonzero functions f and g such that \(g(y) \sin x d x+y^{2} f(x) d y=0\) is exact. Text Transcription: g(y) sin x dx+y^2 f(x) dy=0

Exact Differential Equation Find all nonzero functions g such that \(g(y) e^{y} d x+x y d y=0\) is exact. Text Transcription: g(y) e^y dx+xy dy=0