Referring to the AP Inter 2nd Year Maths Study Material Chapter 9 Differential Equations Exercise 9a Solutions makes it easier to understand complex problems.
AP Inter 2nd Year Maths Differential Equations Solutions Exercise 9a
I.
Question 1.
Determine order and degree of \(\frac{d^4 y}{d x^4}\) + sin(y”’) = 0
Solution:
Given D.E is \(\frac{d^4 y}{d x^4}\) + sin(y”’) = 0 => y”” + sin(y””) = 0 dx
Highest order derivative in the differential equation is y””. Its order is four.
But the D.E is not a polynomial equation in its derivatives. So its degree is not defined.
Question 2.
Determine order and degree of y’+ 5y = 0
Solution:
Given D.E is y’ + 5y = 0
Highest order derivative in the D.E is y’. Its order is one.
It is a polynomial equation in y’. Highest power of y’ is 1. So degree of the D.E is one.
Question 3.
Determine order and degree \(\left(\frac{\mathrm{ds}}{\mathrm{dt}}\right)^4+3 \mathrm{~s} \frac{\mathrm{~d}^2 \mathrm{~s}}{\mathrm{dt}^2}\) = 0
Solution:
Highest order derivative in the given D.E is \(\frac{d^2 s}{d t^2}\). Its order is two.
It is a polynomial equation in \(\frac{d^2 s}{d t^2} \text { and } \frac{d s}{d t}\).
The power of \(\frac{d^2 s}{d t^2}\) is 1. So degree of the D.E is one.
Question 4.
Determine order and degree of \(\left(\frac{d^2 y}{d x^2}\right)^2+\cos \left(\frac{d y}{d x}\right)\) = 0
Solution:
Highest order derivative in the given D.E is \(\frac{d^2 y}{d x^2}\). Its order is 2.
Given differential equation is not a polynomial equation in its derivatives.
Degree of the D.E is not defined.
Question 5.
Determine order and degree of \(\frac{d^2 y}{d x^2}\) = cos3x + sin3x
Solution:
Highest order derivative in the given D.E is \(\frac{d^2 y}{d x^2}\). Its order is two.
It is a polynomial equation in \(\frac{d^2 y}{d x^2}\) and the power is 1. So degree of D.E is 1.
Question 6.
Determine order and degree of (y”’)2 + (y”)3 + (y’)4 + y5 = 0
Solution:
Given D.E is (y”’)2 + (y”)3 + (y’)4 + y5 = 0
Highest order derivative present in the D.E is y”‘. Its order is three.
Given D.E is a polynomial equation in y'”, y” and y’.
Highest powgr raised to y”‘ is 2. So degree of the D.E is 2.
Question 7.
Determine order and degree of y”’ + 2y” + y’= 0
Solution:
GivenD.E is y”’ + 2y’ + y’ = 0
Highest order derivative present in the differential equation is y'” . Its order is 3.
It is a polynomial equation in y'”, y” and y’. The highest power of y”’ is 1. Degree of the D.E is 1.
Question 8.
Determine order and degree of y’ + y = ex
Solution:
Given D.E is y’ +y = ex ⇒ y’ + y – ex = 0
Highest order derivative present in the differential equation is y’. Its order is one.
Given D.E is a polynomial equation in y’ and the highest power of is one.
Degree of the D.E is 1.
Question 9.
Determine order and degree of y” + (y’)2 + 2y = 0
Solution:
Given D.E is y” + (y’)2 + 2y = 0
Highest order derivative present in the differential equation is y” . Its order is two.
Given D.E is a polynomial equation in y” and y’, the highest power of y” is one.
Degree of the D.E is 1.
Question 10.
Determine order and degree of y” + 2y’ + siny = 0
Solution:
Given D.E is y” + 2y’ + siny = 0
Highest order derivative present in the differential equation is y”. Its order is two.
This is a polynomial equation in y” and y’ the highest power of y” is one.
Degree of the D.E is 1