Solution: Use the method discussed under “Equations of the

An equation of the form (18) \(\frac{d y}{d x}=P(x) y^{2}+Q(x) y+R(x)\) is called a generalized Riccati equation. (a) If one solution—say, \(u(x)-\)of (18) is known, show that the substitution \(y=u+1/v\) reduces (18) to a linear equation in ????. (b) Given that \(u(x)=x\) is a solution to \(\frac{d y}{d x}=x^{3}(y-x)^{2}+\frac{y}{x}\), use the result of part (a) to find all the other solutions to this equation. (The particular solution \(u(x)=x\) can be found by inspection or by using a Taylor series method; see Section 8.1.) Equation Transcription: Text Transcription: {dy}over{dx}=P(x)y^{2}+Q(x)y+R(x) u(x)- y=u+1/v u(x)=x \\ {dy}over{dx}=x^{3}(y-x)^{2}+{y}over{x} u(x)=x

Problem 1TWE What properties do solutions to linear equations have that are not shared by solutions to either separable or exact equations? Give some specific examples to support your conclusions.

Problem 1RP In Problems 1–30, solve the equation.

In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form \(y^{\prime}=G(a x+b y)\). \(2 t x\ d x+\left(t^{2}-x^{2}\right) d t=0\) Equation Transcription: Text Transcription: y^{\prime}=G(a x+b y) 2tx dx+(t^{2}-x^{2})dt=0

In Problems 1–30, solve the equation. 2. \(\frac{d y}{d x}-4 y=32 x^{2}\) Equation Transcription: Text Transcription: {dy}over{dx}-4y=32 x^{2}

Problem 3E In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form

Problem 2TWE What properties do solutions to linear equations have that are not shared by solutions to either separable or exact equations? Give some specific examples to support your conclusions.

Problem 3RP In Problems 1–30, solve the equation.

Problem 3TWE Consider the differential equation where a, b, and c are constants. Describe what happens to the asymptotic behavior as x + ? of the solution when the constants a, b, and c are varied. Illustrate with figures and/or graphs.

In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form \(y^{\prime}=G(a x+b y)\). \((t+x+2) d x+(3 t-x-6) d t=0\) Equation Transcription: Text Transcription: y{prime}=G(ax+by) (t+x+2) dx+(3t-x-6)dt=0

Problem 4RP In Problems 1–30, solve the equation.

Problem 5E In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form

Problem 6E In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form

In Problems 1–30, solve the equation. \(2 x y^{3} d x-\left(1-x^{2}\right) d y=0\) Equation Transcription: Text Transcription: 2xy^{3}dx-(1-x^{2})dy=0

Problem 5RP In Problems 1–30, solve the equation.

Problem 7E In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form

Problem 7RP In Problems 1–30, solve the equation.

In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form \(y^{\prime}=G(a x+b y)\). \(\left(y^{3}-\theta y^{2}\right) d \theta+2 \theta^{2} y\ d y=0\) Equation Transcription: Text Transcription: y{\prime}=G(a x+b y) (y^{3}-theta y^{2})d theta+2 theta^{2} y dy=0

In Problems 1–30, solve the equation. \(\frac{dy}{dx}+\frac{2y}x=2x^2y^2\) Equation Transcription: Text Transcription: {dy}over{dx}+{2y}over{x}=2x^2y^2

Use the method discussed under “Homogeneous Equations” to solve Problems 9–16. \(\left(x y+y^{2}\right) d x-x^{2}\ d y=0\) Equation Transcription: Text Transcription: (xy+y^{2})dx-x^{2} dy=0

Problem 10E Use the method discussed under “Homogeneous Equations” to solve Problems 9–16.

Problem 9RP In Problems 1–30, solve the equation.

Problem 11E Use the method discussed under “Homogeneous Equations” to solve Problems 9–16.

Problem 11RP In Problems 1–30, solve the equation.

Problem 10RP In Problems 1–30, solve the equation.

Problem 12E Use the method discussed under “Homogeneous Equations” to solve Problems 9–16.

Problem 12RP In Problems 1–30, solve the equation.

Use the method discussed under “Homogeneous Equations” to solve Problems 9–16. \(\frac{d x}{d t}=\frac{x^{2}+t \sqrt{t^{2}+x^{2}}}{t x}\) Equation Transcription: Text Transcription: {dx}over{dt}={x^{2}+t sqrt{t^{2}+x^{2}}}over{tx}

In Problems 1–30, solve the equation. \(\frac{d y}{d x}-\frac{y}{x}=x^{2} \sin 2 x\) Equation Transcription: Text Transcription: {dy}over{dx}-{y}over{x}=x^{2}sin 2x

Problem 14RP In Problems 1–30, solve the equation.

Use the method discussed under “Homogeneous Equations” to solve Problems 9–16. \(\frac{d y}{d \theta}=\frac{\theta \sec (y / \theta)+y}{\theta}\) Equation Transcription: Text Transcription: {dy}over{d theta}={\theta sec(y/theta)+y}over{theta}

Problem 15E Use the method discussed under “Homogeneous Equations” to solve Problems 9–16.

In Problems 1–30, solve the equation. \(\frac{d y}{d x}=2-\sqrt{2 x-y+3}) Equation Transcription: Text Transcription: {dy}over{dx}=2-sqrt{2x-y+3}

Problem 16E Use the method discussed under “Homogeneous Equations” to solve Problems 9–16.

Problem 16RP In Problems 1–30, solve the equation.

Use the method discussed under “Equations of the Form \(dy/dx=G(ax+by)\)” to solve Problems 17–20. \(d y / d x=\sqrt{x+y}-1\) Equation Transcription: Text Transcription: dy/dx=G(ax+by) dy/dx=sqrt{x+y}-1

Problem 17RP In Problems 1–30, solve the equation.

Use the method discussed under “Equations of the Form \(dy/dx=G(ax+by)\)” to solve Problems 17–20. \(dy/dx=(x+y+2)^2\) Equation Transcription: Text Transcription: dy/dx=G(ax+by) dy/dx=(x+y+2)^2

In Problems 1–30, solve the equation. \(\frac{d y}{d x}=(2 x+y-1)^{2}\) Equation Transcription: Text Transcription: {dy}over{dx}=(2x+y-1)^{2}

Use the method discussed under “Equations of the Form \(dy/dx=G(ax+by)\)” to solve Problems 17–20. \(dy/dx=(x-y+5)^2\) Equation Transcription: Text Transcription: dy/dx=G(ax+by) dy/dx=(x-y+5)^2

In Problems 1–30, solve the equation. \(\left(x^{2}-3 y^{2}\right) d x+2 x y\ d y=0\) Equation Transcription: Text Transcription: (x^{2}-3y^{2}) dx+2xy d y=0

Problem 20RP In Problems 1–30, solve the equation.

Problem 20E Use the method discussed under “Equations of the Form dy/dx = G (ax +by) to solve Problem 17 – 20.

Use the method discussed under “Bernoulli Equations” to solve Problems 21–28. \(\frac{d y}{d x}+\frac{y}{x}=x^{2} y^{2}\) Equation Transcription: Text Transcription: dy over dx+y over x=x^2y^2

Problem 21RP In Problems 1–30, solve the equation.

Problem 22E Use the method discussed under “Bernoulli Equations” to solve Problems 21–28.

Use the method discussed under “Bernoulli Equations” to solve Problems 21–28. \(\frac{d y}{d x}=\frac{2y}{x}-x^{2} y^{2}\) Equation Transcription: Text Transcription: {dy}over{dx}={2y}over{x}-x^{2} y^{2}

Problem 22RP In Problems 1–30, solve the equation.

Problem 23RP In Problems 1–30, solve the equation.

Use the method discussed under “Bernoulli Equations” to solve Problems 21–28. \(\frac{d y}{d x}+\frac{y}{x-2}=5(x-2)y^{1/2}\) Equation Transcription: Text Transcription: {dy}over{dx}+{y}over{x-2}=5(x-2)y^{1/2}

In Problems 1–30, solve the equation. \((\sqrt{y / x}+\cos x) d x+(\sqrt{x / y}+\sin y) d y=0\) Equation Transcription: Text Transcription: (sqrt{y/x}+cos x)dx+(sqrt{x/y}+sin y)dy=0

Use the method discussed under “Bernoulli Equations” to solve Problems 21–28. \(\frac{d x}{d t}+tx^3+\frac{x}{t}=0\) Equation Transcription: Text Transcription: {dx}over{dt}+tx^3+{x}over{t}=0

Problem 25RP In Problems 1–30, solve the equation.

In Problems 1–30, solve the equation. \(\frac{d y}{d x}+x y=0\) Equation Transcription: Text Transcription: {dy}over{dx}+xy=0

In Problems 1–30, solve the equation. \(\frac{d y}{d x}+y=e^{x}y^{-2}\) Equation Transcription: Text Transcription: {dy}over{dx}+y=e^{x}y^{-2}

Problem 27E Use the method discussed under “Bernoulli Equations” to solve Problems 21–28.

In Problems 1–30, solve the equation. \(\) Equation Transcription: Text Transcription: y(x-y-2)dx+x(y-x+4)dy=0

Problem 28E Use the method discussed under “Bernoulli Equations” to solve Problems 21–28.

Use the method discussed under “Equations with Linear Coefficients” to solve Problems 29–32. \((-3x+y-1)dx+(x+y+3)dy=0\) Equation Transcription: Text Transcription: (-3x+y-1)dx+(x+y+3)dy=0

Problem 28RP In Problems 1–30, solve the equation.

Use the method discussed under “Equations with Linear Coefficients” to solve Problems 29–32. \((x+y-1)dx+(y-x-5)dy=0\) Equation Transcription: Text Transcription: (x+y-1)dx+(y-x-5)dy=0

Problem 29RP In Problems 1–30, solve the equation.

Problem 30RP In Problems 1–30, solve the equation.

Use the method discussed under “Equations with Linear Coefficients” to solve Problems 29–32. \((2x-y)dx+(4x+y-3)dy=0\) Equation Transcription: Text Transcription: (2x-y)dx+(4x+y-3)dy=0

Use the method discussed under “Equations with Linear Coefficients” to solve Problems 29–32. \((2x+y+4)dx+(x-2y-2)dy=0\) Equation Transcription: Text Transcription: (2x+y+4)dx+(x-2y-2)dy=0

Problem 32RP In Problems 31–40, solve the initial value problem.

In Problems 31–40, solve the initial value problem. \((x^3-y)dx+x\ dy=0\), \(y(1)=3\) Equation Transcription: Text Transcription: (x^3-y)dx+x dy=0 y(1)=3

In Problems 33–40, solve the equation given in: Problem 1 \(2 t x\ d x+\left(t^{2}-x^{2}\right) d t=0\) Equation Transcription: Text Transcription: 2tx dx+(t^{2}-x^{2})dt=0

Problem 33RP In Problems 31–40, solve the initial value problem.

Problem 34E In Problems 33–40, solve the equation given in: Problem 2.

Problem 34RP In Problems 31–40, solve the initial value problem.

Problem 35RP In Problems 31–40, solve the initial value problem.

In Problems 33–40, solve the equation given in: Problem 3. \(dy/dx+y/x=x^3y^2\) Equation Transcription: Text Transcription: dy/dx+y/x=x^3y^2

Problem 36E In Problems 33–40, solve the equation given in: Problem 4.

In Problems 31–40, solve the initial value problem. \((2x-y)dx+(x+y-3)dy=0\), \(y(0)=2\) Equation Transcription: Text Transcription: (2x-y)dx+(x+y-3)dy=0 y(0)=2

Problem 36RP In Problems 31–40, solve the initial value problem.

In Problems 33–40, solve the equation given in: Problem 6. \(\left(y e^{-2 x}+y^{3}\right) d x-e^{-2 x} d y=0\) Equation Transcription: Text Transcription: (ye^{-2x}+y^{3})dx-e^{-2x}dy=0

In Problems 31–40, solve the initial value problem. \(\sqrt{y}\ d x+\left(x^{2}+4\right) d y=0\), \(y(0)=4\) Equation Transcription: Text Transcription: \sqrt{y} d x+\left(x^{2}+4\right) d y=0 y(0)=4

In Problems 33–40, solve the equation given in: Problem 7. \(\cos(x+y)dy=\sin(x+y)dx\) Equation Transcription: Text Transcription: cos(x+y)dy=sin(x+y)dx

In Problems 31–40, solve the initial value problem. \(\frac{d y}{d x}-\frac{2 y}{x}=x^{-1} y^{-1}\), \(y(1)=3\) Equation Transcription: Text Transcription: {dy}over{dx}-{2y}over{x}=x^{-1} y^{-1} y(1)=3

Problem 40RP In Problems 31–40, solve the initial value problem.

Use the substitution \(v=x-y+2\) to solve equation (8). Equation Transcription: Text Transcription: v=x-y+2

In Problems 33–40, solve the equation given in: Problem 8. \(\left(y^{3}-\theta y^{2}\right) d \theta+2 \theta^{2} y\ d y=0\) Equation Transcription: Text Transcription: (y^{3}-theta y^{2}) d theta+2 theta^{2}y dy=0

Problem 41RP Express the solution to the following initial value problem using a definite integral:

Problem 42E Use the substitution y = ux2 to solve

Problem 43E (a) Show that the equation dy/dx = f (x,y) is homogeneous if and only if f (tx, ty) = f (x,y). [hint: let t = 1/x.] (b) A function H(x,y) is called homogeneous of order n if H(tx,ty) = tn H (x,y). Show that the equation is homogeneous if M (x,y) and N (x,y) are both homogeneous of the same order

Show that equation (13) reduces to an equation of the form \(\frac{d y}{d x}=G(a x+b y) \), when \(a_{1} b_{2}=a_{2} b_{1}\). [Hint: If \(a_{1} b_{2}=a_{2} b_{1}\), then \(a_{2} / a_{1}=b_{2} / b_{1}=k\), so that \(a_{2}=k a_{1}\) and \(b_{2}=k b_{1}\).] Equation Transcription: Text Transcription: {dy}over{dx}=G(ax+by) a_{1}b_{2}=a_{2}b_{1} a_{1}b_{2}=a_{2}b_{1} a_{2}/a_{1}=b_{2}/b_{1}=k a_{2}=ka_{1} b_{2}=kb_{1}

Problem 46E Magnetic Field Lines. As described in Problem 20 of Exercises 1.3, the magnetic field lines of a dipole satisfy Solve this equation and sketch several of these lines.

Coupled Equations. In analyzing coupled equations of the form \(\frac{d y}{d t}=a x+b y\), \(\frac{d x}{d t}=\alpha x+\beta y\), where \(a,\ b,\ \alpha\) and \(\beta\) are constants, we may wish to determine the relationship between ???? and ???? rather than the individual solutions \(x(t),\ y(t)\). For this purpose, divide the first equation by the second to obtain (17) \(\frac{d y}{d x}=\frac{a x+b y}{\alpha x+\beta y}\). This new equation is homogeneous, so we can solve it via the substitution \(v=y / x\). We refer to the solutions of (17) as integral curves. Determine the integral curves for the system \(\frac{d y}{d t}=-4 x-y\), \(\frac{d x}{d t}=2 x-y\). Equation Transcription: Text Transcription: {dy}over{dt}=ax+by {dx}over{dt}=alpha x+beta y a, b, alpha beta x(t), y(t) {dy}over{dx}={ax+by}over{alpha x+beta y} v=y/x {dy}over{dt}=-4 x-y {dx}over{dt}=2 x-y

In Problems 1–8, identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form \(y^{\prime}=G(a x+b y)\). \((y-4x-1)^{2}\ dx-dy=0\) Equation Transcription: Text Transcription: (y'=G(ax+by) (y-4x-1)^{2}dx-dy=0

In Problems 33–40, solve the equation given in: Problem 5.