Which of the given differential equations are linear and to determine the order of each equation.

Question

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Answer 1

Question

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

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According to Newton's law of universal gravitation, the force F between two bodies of constant mass GmM m and M is given by the formula F = , where G is the gravitational constant and d is the d² distance between the bodies. a. Suppose that G, m, and M are constants. Find the rate of change of force F with respect to distance d. F' (d) 2GmM b. Find the rate of change of force F with gravitational constant G = 6.67 × 10-¹¹ Nm²/kg², on two bodies 5 meters apart, each with a mass of 250 kilograms. Answer in scientific notation, rounding to 2 decimal places. -6.67x10 N/m syntax incomplete.

Answer 2

Question

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

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Answer 3

Question

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

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Answer 4

Question

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

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Answer 5

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Answer 6

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Answer 7

Chapter 9, Problem 1CRE

To determine

Which of the given differential equations are linear and to determine the order of each equation.

Answer 8

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

(b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Initially check for linearity. Then observe the higher order derivative to determine the order of the equation.

Given:

(a) $y' = y^{5} - 3 x^{4} y$

(b) $y' = x^{5} - 3 x^{4} y$

(c) $y = y''' - 3 x \sqrt{y}$

(d) $\sin x . y'' = y - 1$

Calculation:

(a) $y' = y^{5} - 3 x^{4} y$ - $y^{5}$ is a non-linear term, hence this is not a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(b) $y' = x^{5} - 3 x^{4} y$ - this is a linear equation. This is a first order equation as the first derivative is the highest order derivative here.

(c) $y = y''' - 3 x \sqrt{y}$ - $\sqrt{y}$ is a non-linear term so this is not a linear equation. This is a third order equation as the third derivative is the highest order derivative here.

(d) $\sin x . y'' = y - 1$ - this is a linear equation. This is a second order equation as the second derivative is the highest order derivative here.

Conclusion:

Hence, (b) and (d) are linear equations with first and second order respectively while (a) and (c) are non-linear equations with first and third order respectively.

Answer 12

Ch. 9.1 - Prob. 1PQ Ch. 9.1 - Prob. 2PQ Ch. 9.1 - Prob. 3PQ Ch. 9.1 - Prob. 4PQ Ch. 9.1 - Prob. 5PQ Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E

Which of the given differential equations are linear and to determine the order of each equation.

Which of the given differential equations are linear and to determine the order of each equation.

Answer to Problem 1CRE

Explanation of Solution

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Chapter 9 Solutions