 QUESTION

# Determine the order and degree (if defined) of the differential equations given in the following:1. $\dfrac{{{d^4}y}}{{d{x^4}}} + \sin ({y^{'''}}) = 0$ 2. $y' + 5y = 0$ 3. ${\left( {\dfrac{{ds}}{{dt}}} \right)^4} + 3s\dfrac{{{d^2}s}}{{d{t^2}}} = 0$ 4. ${\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} - \cos \left( {\dfrac{{dy}}{{dx}}} \right) = 0$5. $\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right) = \cos 3x + \sin 3x$ 6. ${\left( {{y^{'''}}} \right)^2} + {\left( y \right)^3} + {\left( y \right)^4} + {y^5} = 0$ 7. $y + 2{y^{''}} + y = 0$ 8. $y' + y = {e^x}$ 9. ${y^{''}} + (y)^2 + \sin y = 0$ 10.${y^{''}} + 2y + \sin y = 0$

Hint: Let's make use of the definition of order and degree of a derivative and lets try
to solve this problem.
Order of a differential equation refers to the highest numbered derivative in the equation and degree refers to the power to which the highest numbered derivative is raised.

Now let us solve the equations give
1. $\dfrac{{{d^4}y}}{{d{x^4}}} + \sin ({y^{'''}}) = 0$
Ans: In this case, the highest numbered derivative is 4 and it is raised to power of 1.
So, Order=4
Degree=1

2. $y' + 5y = 0$
Ans: In this case, the highest order derivative is 1 and it is raised to the power of 1
So, Order=1
Degree=1

3. ${\left( {\dfrac{{ds}}{{dt}}} \right)^4} + 3s\dfrac{{{d^2}s}}{{d{t^2}}} = 0$
Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1
So, Order=2
Degree=1

4. ${\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} - \cos \left( {\dfrac{{dy}}{{dx}}} \right) = 0$
Ans: In this case,the highest order derivative is 2 and it is raised to the power of 2
So, Order=2
Degree=2

5. $\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right) = \cos 3x + \sin 3x$
Ans: In this case,the highest order derivative is 2 and it is raised to the power of 1
So, Order=2
Degree=1

6. ${\left( {{y^{'''}}} \right)^2} + {\left( y \right)^3} + {\left( y \right)^4} + {y^5} = 0$
Ans: In this case, the highest order derivative is 3 and it is raised to the power of 2
So, Order=3
Degree=2

7. $y + 2{y^{''}} + y = 0$
Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1
So, Order=2
Degree=1

8. $y' + y = {e^x}$
Ans: In this case, the highest order derivative is 1 and it is raised to the power of 1
So, Order=1
Degree=1

9. ${y^{''}} + (y)^2 + \sin y = 0$
Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1
So, Order=2
Degree=1

10. ${y^{''}} + 2y + \sin y = 0$
Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1
So, Order=2
Degree=1

Note: In these types of questions it has to be noted that the order of the differential equation is the highest order derivative and not the highest power in the equation.