Class 12 NCERT Solutions- Mathematics Part ii – Chapter 9– Differential Equations Exercise 9.1
Order of a Differential Equation
Order of a differential equation is defined as the highest order derivative present in the equation. It indicates the complexity of the equation and the number of independent variables involved.
Degree of a Differential Equation
Degree of a differential equation is defined for polynomial differential equations. It refers to the highest power of the highest order derivative present in the equation when it is expressed in its polynomial form.
Determine the order and degree (if defined) of differential equations given in Exercises 1 to 10.
1. [Tex]\frac{d^4y}{dx^4}+sin(y”’) = 0[/Tex]
Solution:
The highest derivative present is d4y/dx4. Therefore, the order of this differential equation is 4.
As the equation is not in the polynomial form, thus it’s degree is not defined.
2. y’ + 5y = 0
Solution:
The highest derivative present is y’. Therefore, the order of this differential equation is 1.
It is a polynomial equation in y’. The highest power raised to y’ is 1. Hence, its degree is 1.
3. [Tex][\frac{ds}{dt}]^4+3s\frac{d^2s}{dt^2}=0[/Tex]
Solution:
The highest derivative present is d2s/dt2. Therefore, the order of this differential equation is 2.
It is a polynomial equation in d2s/dt2 and ds/dt. The highest power raised to d2s/dt2 is 1. Hence, its degree is 1.
4. [Tex][\frac{d^2y}{dx^2}]^2+cos[\frac{dy}{dx}]=0[/Tex]
Solution:
The highest derivative present is d2y/dx2. Therefore, the order of this differential equation is 2.
As the equation is not in the polynomial form, thus it’s degree is not defined.
5. [Tex]\frac{d^2y}{dx^2}=cos3x+sin3x[/Tex]
Solution:
The highest derivative present is d2y/dx2. Therefore, the order of this differential equation is 2.
It is a polynomial equation in d2y/dx2. The highest power raised to it is 1. Hence, its degree is 1.
6. (y”’)2 + (y”)3 + (y’)4 + y5 = 0
Solution:
The highest derivative present is y”’. Therefore, the order of this differential equation is 3.
It is a polynomial equation in y”’, y”, and y’. The highest power raised to y’” is 2. Hence, its degree is 2.
7. y”’ + 2y” + y’ = 0
Solution:
The highest derivative present is y”’. Therefore, the order of this differential equation is 3.
It is a polynomial equation in y”’, y”, and y’. The highest power raised to y’” is 1. Hence, its degree is 1.
8. y’ + y = ex
Solution:
The highest derivative present is y’. Therefore, the order of this differential equation is 1.
It is a polynomial equation in y’. The highest power raised to y’ is 1. Hence, its degree is 1.
9. y” + (y’)2 + 2y = 0
Solution:
The highest derivative present is y”. Therefore, the order of this differential equation is 2.
It is a polynomial equation in y” and y’. The highest power raised to y’’ is 1. Hence, its degree is 1.
10. y” + 2y’ + sin y = 0
Solution:
The highest derivative present is y”. Therefore, the order of this differential equation is 2.
It is a polynomial equation in y” and y’. The highest power raised to y’’ is 1. Hence, its degree is 1.
11. The degree of the differential equation [Tex](\frac{d^2y}{dx^2})^3+(\frac{dy}{dx})^2+ \sin(\frac{dy}{dx})+1=0[/Tex] is:
(A) 3 (B) 2 (C) 1 (D) Not Defined
Solution:
As the given equation is not represented as polynomial, thus it’s degree is not defined. Thus, option D is correct.
12. The order of the differential equation [Tex]2x^2\frac{d^2y}{dx^2}-3\frac{dy}{dx}+y=0[/Tex] is:
(A) 2 (B) 1 (C) 0 (D) Not Defined
Solution:
The highest derivative present is d2y/dx2. Therefore, the order of this differential equation is 2.
Thus, the correct answer is Option A.
Contact Us