Chapter 9, Problem 1CRE

To determine

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

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) $y\text{'}=y^5-3x^{4}y$

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

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

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

Calculation:

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

(c) $y=y\text{'}\text{'}\text{'}-3x\sqrt{y}$ - $\sqrt{y}$ 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) $\sin x.y\text{'}\text{'}=y-1$ - 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.