**Q.1: Find the degree and order of the differential equation \(\frac{\mathrm{d} ^{4}y}{\mathrm{d} x^{4}}\)+sin (y ^{m}) = 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:**

\(\frac{d^{2}s}{dt^{2}}\) 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:**

\(\\\frac{d^{2}y}{dx^{2}}\) **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:**

**\(\frac{d^{2}y}{dx^{2}}\) 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}+ y^{5} = 0**

** **

**Solution: ****(y”’) ^{2} + (y”)^{3} + (y’)^{4}+ y^{5} = 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 = e ^{x}**

** **

**Solution: ****y’ + y = e ^{x}**

\(\boldsymbol{\Rightarrow }\) y’ + y – e^{x} = 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:**

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).**