Q.1: Find the degree and order of the differential equation \(\frac{\mathrm{d} ^{4}y}{\mathrm{d} x^{4}}\)+sin (ym) = 0.
Solution:
\(\frac{\mathrm{d} ^{4}y}{\mathrm{d} x^{4}}\) + sin (y”’) = 0
y”” + sin (y”’) = 0
y”” is the highest order derivative present in the differential equation.
Therefore, the order is four.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
Q.2: Find the degree and order of differential equation y’ + 5y = 0
Solution:
Given: y’ + 5y = 0
y’ is the highest order derivative present in the differential equation.
Therefore, the order is one.
The given differential equation is a polynomial equation in y’. The highest degree derivative present in the differential equation is y’.
Therefore, the degree is one.
Q.3: Find the degree and order of differential equation \((\frac{\mathrm{d} s}{\mathrm{d} t})^{4}+3s\frac{\mathrm{d}^{2} s}{\mathrm{d} t}=0\).
Solution:
\((\frac{\mathrm{d} s}{\mathrm{d} t})^{4}+3s\frac{\mathrm{d}^{2} s}{\mathrm{d} t}=0\)\(\frac{d^{2}s}{dt^{2}}\) is the highest order derivative present in the differential equation.
Therefore, the order is two.
The given differential equation is a polynomial equation in \(\frac{d^{2}s}{dt^{2}}\) and \(\frac{ds}{dt}\).The power raised to \(\frac{d^{2}s}{dt^{2}}\) is 1.
Hence, its degree is one.
Q.4: Find the degree and order of differential equation \((\frac{d^{2}y}{dx^{2}})^{2}+cos(\frac{dy}{dx})=0\).
Solution:
\((\frac{d^{2}y}{dx^{2}})^{2}+cos(\frac{dy}{dx})=0\)\(\\\frac{d^{2}y}{dx^{2}}\) is the highest order derivative present in the differential equation.
Therefore, the order is two.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
Q.5: Find the degree and order of differential equation \(\frac{d^{2}y}{dx^{2}} =\cos 3x+\sin 3x\)
Solution:
\(\frac{d^{2}y}{dx^{2}} =\cos 3x+\sin 3x\\ \\ \Rightarrow \frac{d^{2}y}{dx^{2}}-\cos 3x-\sin 3x=0\)\(\frac{d^{2}y}{dx^{2}}\) is the highest order derivative present in the differential equation.
Therefore, the order is two.
It is a polynomial equation in \(\frac{d^{2}y}{dx^{2}}\) and the power raised to \(\frac{d^{2}y}{dx^{2}}\) is 1.
Hence, its degree is one.
Q.6: Find the degree and order of differential equation (y”’)2 + (y”)3 + (y’)4+ y5 = 0
Solution: (y”’)2 + (y”)3 + (y’)4+ y5 = 0
y”’ is the highest order derivative present in the differential equation.
Therefore, the order is three.
It is a polynomial equation in y”’, y” and y’.
The power of y”’ is 2.
Hence, its degree is 2.
Q.7: Find the degree and order of differential equation y”’ + 2y” + y’ = 0.
Solution:
Given: y”’ + 2y” + y’ = 0.
y”’ is the highest order derivative present in the differential equation.
Therefore, the order is three.
It is a polynomial equation in y”’, y”, and y’.
The power of y”’ is 1.
Hence, its degree is 1.
Q.8: Find the degree and order of differential equation y’ + y = ex
Solution: y’ + y = ex
\(\boldsymbol{\Rightarrow }\) y’ + y – ex = 0
y’ is the highest order derivative present in the differential equation.
Therefore, the order is one.
It is a polynomial equation in y’.
The power raised to y”’ is 1.
Hence, its degree is 1.
Q.9: Find the degree and order of differential equation y” + (y’)2 +2y = 0.
Solution:
y” + (y’)2 +2y = 0
y” is the highest order derivative present in the differential equation.
Therefore, the order is two.
It is a polynomial equation in y” + y’.
The power raised to y” is 1.
Hence, its degree is 1.
Q.10: The degree of differential equation \((\frac{d^{2}y}{dx^{2}})+(\frac{dy}{dx})^{2}+\sin (\frac{dy}{dx})+1=0\) is:
(i) 3
(ii) 2
(iii) 1
(iv) not defined
Solution:
\((\frac{d^{2}y}{dx^{2}})+(\frac{dy}{dx})^{2}+\sin (\frac{dy}{dx})+1=0\)The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
Hence, the answer is (iv).
Q.11: The degree of differential equation: \(2x^{2}\;\frac{d^{2}y}{dx^{2}}-3\frac{dy}{dx}+y=0\)is:
(i) 2
(ii) 1
(iii) 0
(iv) not defined
Solution:
\(\frac{d^{2}y}{dx^{2}}\) is the highest order derivative present in the differential equation.
Therefore, the order is two.
Hence, the correct answer is (i).
