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Question: Determine the order and degree (if defined) of the differential equations given in the following: ...

Determine the order and degree (if defined) of the differential equations given in the following:
1. d4ydx4+sin⁡(y′′′)=0\dfrac{{{d^4}y}}{{d{x^4}}} + \sin ({y^{'''}}) = 0
2. y′+5y=0y' + 5y = 0
3. (dsdt)4+3sd2sdt2=0{\left( {\dfrac{{ds}}{{dt}}} \right)^4} + 3s\dfrac{{{d^2}s}}{{d{t^2}}} = 0
4. (d2ydx2)2−cos⁡(dydx)=0{\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} - \cos \left( {\dfrac{{dy}}{{dx}}} \right) = 0
5. (d2ydx2)=cos⁡3x+sin⁡3x\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right) = \cos 3x + \sin 3x
6. (y′′′)2+(y)3+(y)4+y5=0{\left( {{y^{'''}}} \right)^2} + {\left( y \right)^3} + {\left( y \right)^4} + {y^5} = 0
7. y+2y′′+y=0y + 2{y^{''}} + y = 0
8. y′+y=exy' + y = {e^x}
9. y′′+(y)2+sin⁡y=0{y^{''}} + (y)^2 + \sin y = 0
10.y′′+2y+sin⁡y=0{y^{''}} + 2y + \sin y = 0

Explanation

Solution

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. d4ydx4+sin⁡(y′′′)=0\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=0y' + 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. (dsdt)4+3sd2sdt2=0{\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. (d2ydx2)2−cos⁡(dydx)=0{\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. (d2ydx2)=cos⁡3x+sin⁡3x\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. (y′′′)2+(y)3+(y)4+y5=0{\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+2y′′+y=0y + 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=exy' + 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{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{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.