Chapter 10, Problem 1CRE

a.

To determine

To find:Whether the differential equation $y\text{'}=y^5-3x^{4}y$ is first-order linear.

Answer to Problem 1CRE

The equation $y\text{'}=y^5-3x^{4}y$ is non-line first-order.

Explanation of Solution

Given information:The differential equation is $y\text{'}=y^5-3x^{4}y$ .

Definition used:The order of a differential equation is the order of the highest derivative appearing in the equation.

A differential equation is called linear if it can be written in the form

  $a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+a_1(x)y^{\text{'}}+a_0(x)y=b(x)$

The linear equation cannot have terms such as $y^3,yy\text{'},\sin y.$

Calculation:

  $y\text{'}=y^5-3x^{4}y$(Given)

The given equation contains the term $y^5$ , therefore it is not linear.

The order of the equation is 1.

Hence, the equation $y\text{'}=y^5-3x^{4}y$ is non-linear first-order.

b.

To determine

To find: Whether the differential equation $y\text{'}=x^5-3x^{4}y$ is first-order linear.

Answer to Problem 1CRE

The equation $y\text{'}=x^5-3x^{4}y$ is first-order linear.

Explanation of Solution

Given information: The differential equation is $y\text{'}=x^5-3x^{4}y$ .

Definition used: The order of a differential equation is the order of the highest derivative appearing in the equation.

A differential equation is called linear if it can be written in the form

  $a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+a_1(x)y^{\text{'}}+a_0(x)y=b(x)$

The linear equation cannot have terms such as $y^3,yy\text{'},\sin y.$

Calculation:

  $y\text{'}=x^5-3x^{4}y$(Given)

The given equation is a linear.

The order of the equation is 1.

Hence, the equation $y\text{'}=x^5-3x^{4}y$ is first-order linear equation.

c.

To determine

To find: Whether the differential equation $y=y\text{'}-3x\sqrt{y}$ is first-order linear.

Answer to Problem 1CRE

The equation $y=y\text{'}-3x\sqrt{y}$ is first-order non-linear.

Explanation of Solution

Given information: The differential equation is $y=y\text{'}-3x\sqrt{y}$ .

Definition used: The order of a differential equation is the order of the highest derivative appearing in the equation.

A differential equation is called linear if it can be written in the form

  $a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+a_1(x)y^{\text{'}}+a_0(x)y=b(x)$

The linear equation cannot have terms such as $y^3,yy\text{'},\sin y.$

Calculation:

  $y=y\text{'}-3x\sqrt{y}$(Given)

The given equation is non-linear as it contains the term $\sqrt{y}$ .

The order of the equation is 1.

Hence, the equation $y=y\text{'}-3x\sqrt{y}$ is first-order non-linear equation.

d.

To determine

To find: Whether the differential equation $\sin x.y\text{'}=y-1$ is first-order linear.

Answer to Problem 1CRE

The equation $\sin xy\text{'}=y-1$ is first-order linear.

Explanation of Solution

Given information: The differential equation is $\sin xy\text{'}=y-1$ .

Definition used: The order of a differential equation is the order of the highest derivative appearing in the equation.

A differential equation is called linear if it can be written in the form

  $a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+a_1(x)y^{\text{'}}+a_0(x)y=b(x)$

The linear equation cannot have terms such as $y^3,yy\text{'},\sin y.$

Calculation:

  $\sin x\cdot y\text{'}=y-1$(Given)

The given equation is a linear.

The order of the equation is 1.

Hence, the equation $\sin x\cdot y\text{'}=y-1$ is first-order linear equation.