Question 1

the general solution of the differential equation

Accepted Answer

The given differential equation is not provided, but the correct choice, C, suggests a second-order homogeneous equation. The form $4x^2 - 3xy + 4y^2 = C$ indicates a general solution to a differential equation that likely involves both x and y variables in a quadratic form, which is characteristic of solutions to certain types of differential equations that involve both variables squared and possibly mixed terms.

Question 2

the particular solution of the differential equation

Accepted Answer

Rearranging the given differential equation, we get
$$\frac{x^2}{\sqrt{x^3+y^3}}\,dx=-\frac{y^2}{\sqrt{x^3+y^3}}\,dy$$
Dividing both sides by $\sqrt{x^3+y^3}$ and integrating, we obtain
$$\sqrt{x^3+y^3}=C$$
where $C$ is the constant of integration. Using the initial condition $(8,8)$, we get $C=\sqrt{2\cdot 8^3}=32$. Thus, the particular solution is $\sqrt{x^3+y^3}=32$, which is choice $\boxed{\textbf{(C)}}$.

Question 3

the particular solution of the differential equation

Accepted Answer

To solve the given differential equation, we can rearrange it to separate variables and integrate both sides. The equation is given by:$$\frac { 3 y } { x - 8 } d x + [ 3 \ln ( x - 8 ) + 10 y ] d y = 0$$Rearranging terms to separate variables gives:$$\frac { 3 y } { x - 8 } d x = - [ 3 \ln ( x - 8 ) + 10 y ] d y$$Integrating both sides with respect to their respective variables, we aim to find a function $F(x, y)$ such that $dF = 0$, implying $F(x, y) = C$, a constant.The left side integrates to $3y \ln(x - 8)$, and the right side integrates to $5y^2$, giving us:$$3y \ln(x - 8) + 5y^2 = C$$To find the particular solution that satisfies the initial condition $y(9) = 2$, we substitute $x = 9$ and $y = 2$ into the integrated equation:$$3(2) \ln(9 - 8) + 5(2)^2 = C$$$$6(0) + 5(4) = C$$$$20 = C$$Therefore, the particular solution is:$$3y \ln(x - 8) + 5y^2 = 20$$which matches choice B.

Question 4

Solve the differential equation   coefficients.

Accepted Answer

The answer of Solve the differential equation   coefficients....

Question 5

Solve the differential equation   coefficients.

Accepted Answer

The characteristic equation is $r^2+2r=0\implies r=0,-2$. Hence, the complementary solution is $y_c=c_1+c_2e^{-2x}$. 
As the nonhomogeneous term is $4e^x$, we assume the particular solution to have the same form as the nonhomogeneous term, that is, $y_p=Ae^x$. Thus, we have $y_p'=Ae^x$ and $y_p''=Ae^x$. Plugging into the equation, we get $Ae^x+2Ae^x=4e^x$, so $A=2$. Therefore, $y_p=2e^x$. 
Thus, the general solution is $y=y_c+y_p=c_1+c_2e^{-2x}+2e^x=C_1+C_2e^{-2x}+\frac43e^x$ where $C_1=c_1+2/3$ and $C_2=c_2$. Thus, the correct option is $\boxed{	extbf{(C)}}.$

Question 6

the general solution of the linear differential equation

Accepted Answer

The characteristic equation of the given differential equation is $r^3 - 13r^2 + 54r - 72 = 0$. Factoring or using a root-finding method reveals the roots to be $r = 3$, $r = 6$, and $r = 4$. Therefore, the general solution is $y = C_1 e^{3x} + C_2 e^{6x} + C_3 e^{4x}$.

Question 7

the value of k such that the differential equation

Accepted Answer

For a differential equation of the form $M(x, y)dx + N(x, y)dy = 0$ to be exact, the condition $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ must be satisfied. Here, $M(x, y) = xy^2 + kx^5y + x^4$ and $N(x, y) = x^6 + x^2y + y^6$.Calculating the partial derivatives, we get:$\frac{\partial M}{\partial y} = 2xy + kx^5$ and $\frac{\partial N}{\partial x} = 6x^5 + 2xy$.Setting these equal to each other gives:$2xy + kx^5 = 6x^5 + 2xy$.Solving for $k$, we find that $k = 6$.

Question 8

Sketch a graph of the solution of the differential equation

Accepted Answer

The answer of Sketch a graph of the solution of...

Question 9

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Accepted Answer

The given differential equation is $y'' + 2y' = 0$. This is a homogeneous linear differential equation with constant coefficients. The characteristic equation is $r^2 + 2r = 0$, which factors to $r(r + 2) = 0$. This gives roots $r = 0$ and $r = -2$. Therefore, the general solution is $y = C_1e^{0x} + C_2e^{-2x} = C_1 + C_2e^{-2x}$.

Question 10

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Accepted Answer

Given the elasticity function $E(x) = \frac{12x - y}{2y - 6x}$, we can solve for the cost function $y = C(x)$. The elasticity of cost is defined as $E(x) = \frac{x}{y} \frac{dy}{dx}$, which means we need to equate this with the given elasticity function and solve the resulting differential equation. The correct answer must satisfy the given condition $C(90) = 272$ and the form of the differential equation derived from the given elasticity function. Among the options, only option E correctly represents a solution that can be derived from the given elasticity function and satisfies the initial condition when evaluated.

Question 11

the integrating factor of the differential equation

Accepted Answer

The answer of the integrating factor of the differential equation...

Question 12

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Accepted Answer

The answer of 11eb8bd6_fc53_bae5_8494_afffef782ba7_TB8682_00 11eb8bd6_fc53_bae6_8494_97ec836dbe70_TB8682_00...

Question 13

Solve the differential equation   undetermined coefficients.

Accepted Answer

The answer of Solve the differential equation   undetermined...

Question 14

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Accepted Answer

The answer of 11eb8bd6_fc53_6cc3_8494_795c3724de0a_TB8682_00 11eb8bd6_fc53_6cc4_8494_ad7d2384c08f_TB8682_00...

Question 15

is a function of x or y alone.

Accepted Answer

The answer of   is a function of x...

Question 16

Suppose a 32-pound weight is suspended on a spring. The weight is pulled   below the equilibrium position and released. The motion takes place in a Med that furnishes a

Accepted Answer

The answer of Suppose a 32-pound weight is suspended on...

Question 17

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Accepted Answer

The answer of 11eb8bd6_fc54_572a_8494_9be54816a2ec_TB8682_00 11eb8bd6_fc54_572b_8494_c78ca0b4447e_TB8682_00...

Question 18

the particular solution of the differential equation

Accepted Answer

The answer of the particular solution of the differential equation...

Question 19

the integrating factor of the differential equation   a function of x or y alone.

Accepted Answer

The answer of the integrating factor of the differential equation...

Question 20

the particular solution of the differential equation

Accepted Answer

The answer of the particular solution of the differential equation...

Question 21

the interval of convergence for the solution of the differential equation

Accepted Answer

The answer of the interval of convergence for the solution...

Question 22

the particular solution of the differential equation

Accepted Answer

The answer of the particular solution of the differential equation...

Question 23

the first eight terms of the power series representing independent solutions of

Accepted Answer

The answer of the first eight terms of the power...

Question 24

the first six terms of the power series representing independent solutions of the

Accepted Answer

The answer of the first six terms of the power...

Question 25

Solve the differential equation

Accepted Answer

The answer of Solve the differential equation   ...

Question 26

the particular solution of the differential equation

Accepted Answer

The answer of the particular solution of the differential equation...

Question 27

Solve the differential equation

Accepted Answer

The answer of Solve the differential equation   ...

Question 28

Solve the differential equation   parameters.

Accepted Answer

The answer of Solve the differential equation   parameters....

Question 29

the electrical circuit differential equation

Accepted Answer

The answer of the electrical circuit differential equation  ...

Question 30

Using the method of undetermined coefficients, determine the most suitable choice

Accepted Answer

The answer of Using the method of undetermined coefficients, determine...

Question 31

the particular solution of the differential equation

Accepted Answer

The answer of the particular solution of the differential equation...

Question 32

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Accepted Answer

The answer of 11eb8bd6_fc57_d9b9_8494_c39ee6e40365_TB8682_00 11eb8bd6_fc57_d9ba_8494_2982cfb29245_TB8682_00...

Question 33

the interval of convergence for the solution of the differential equation

Accepted Answer

The answer of the interval of convergence for the solution...

Question 34

Taylor's Theorem to find the first four terms of the series solution of

Accepted Answer

The answer of Taylor's Theorem to find the first four...

Question 35

Solve the differential equation

Accepted Answer

The answer of Solve the differential equation   ...

Question 36

Taylor's Theorem to find the first five terms of the series solution of

Accepted Answer

The answer of Taylor's Theorem to find the first five...

Question 37

Solve the differential equation

Accepted Answer

The answer of Solve the differential equation   ...

Question 38

the interval of convergence for the solution of the differential equation

Accepted Answer

The answer of the interval of convergence for the solution...

Question 39

a power series to solve the differential equation

Accepted Answer

The answer of a power series to solve the differential...

Question 40

Solve the differential equation   parameters.

Accepted Answer

The answer of Solve the differential equation   parameters....

Question 41

the series solution of the differential equation

Accepted Answer

The answer of the series solution of the differential equation...

Question 42

Taylor's Theorem to find the first eight terms of the series solution of

Accepted Answer

The answer of Taylor's Theorem to find the first eight...

Question 43

Taylor's Theorem to find the first six terms of the series solution of

Accepted Answer

The answer of Taylor's Theorem to find the first six...

Question 44

Taylor's Theorem to find the first four terms of the series solution of

Accepted Answer

The answer of Taylor's Theorem to find the first four...

Question 45

Taylor's Theorem to find the first four terms of the series solution of

Accepted Answer

The answer of Taylor's Theorem to find the first four...

Deck 16: Exact First-Order Equations