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.${y^{''}} + 2y + \sin y = 0$

Answer 1

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.

Complete step-by-step answer:
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.