## Chapter 9 Differential Equations EXERCISE 9.1

### EXERCISE 9.1

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10 .

**1.** $\frac{d^{4} y}{d x^{4}}+\sin (y^{\prime \prime \prime})=0$

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**Solution**

$\Rightarrow y^{\prime \prime \prime \prime}+\sin (y^{\prime \prime \prime})=0$

The highest order derivative present in the differential equation is $y^{\prime \prime \prime \prime}$. Therefore, its order is four.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

**2.** $y^{\prime}+5 y=0$

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**Solution**

The given differential equation is:

$y^{\prime}+5 y=0$

The highest order derivative present in the differential equation is $y^{\prime}$. Therefore, its order is one.

It is a polynomial equation in $y^{\prime}$. The highest power raised to $y^{\prime}$ is 1 . Hence, its degree is one.

**3.** $(\frac{d s}{d t})^{4}+3 s \frac{d^{2} s}{d t^{2}}=0$

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**Solution**

$(\frac{d s}{d t})^{4}+3 \frac{d^{2} s}{d t^{2}}=0$

The highest order derivative present in the given differential equation is $\frac{d^{2} s}{d t^{2}}$. Therefore, its order is two.

It is a polynomial equation in $\frac{d^{2} s}{d t^{2}}$ and $\frac{d s}{d t}$. The power raised to $\frac{d^{2} s}{d t^{2}}$ is 1 . Hence, its degree is one.

**4.** $(\frac{d^{2} y}{d x^{2}})^{2}+\cos (\frac{d y}{d x})=0$

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**Solution**

$(\frac{d^{2} y}{d x^{2}})^{2}+\cos (\frac{d y}{d x})=0$

The highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, its order is 2.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

**5.** $\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x$

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**Solution**

$\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x$

$\Rightarrow \frac{d^{2} y}{d x^{2}}-\cos 3 x-\sin 3 x=0$

The highest order derivative present in the differential equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, its order is two.

It is a polynomial equation in $\frac{d^{2} y}{d x^{2}}$ and the power raised to $\frac{d^{2} y}{d x^{2}}$ is 1 .

Hence, its degree is one.

**6.** $(y^{\prime \prime \prime})^{2}+(y^{\prime \prime})^{3}+(y^{\prime})^{4}+y^{5}=0$

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**Solution**

$(y^{\prime \prime \prime})^{2}+(y^{\prime \prime})^{3}+(y^{\prime})+y^{5}=0$

The highest order derivative present in the differential equation is $y^{\prime \prime \prime}$. Therefore, its order is three.

The given differential equation is a polynomial equation in $y^{\prime \prime \prime}, y^{\prime \prime}$, and $y^{\prime}$.

The highest power raised to $y^{\prime \prime \prime}$ is 2 . Hence, its degree is 2 .

**7.** $y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$

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**Solution**

$y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$

The highest order derivative present in the differential equation is $y^{\prime \prime \prime}$. Therefore, its order is three.

It is a polynomial equation in $y^{\prime \prime \prime}, y^{\prime \prime}$ and $y^{\prime}$. The highest power raised to $y^{\prime \prime \prime}$ is 1 . Hence, its degree is 1 .

**8.** $y^{\prime}+y=e^{x}$

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**Solution**

$ \begin{aligned} & y^{\prime}+y=e^{x} \\ & \Rightarrow y^{\prime}+y-e^{x}=0 \end{aligned} $

The highest order derivative present in the differential equation is $y^{\prime}$. Therefore, its order is one.

The given differential equation is a polynomial equation in $y^{\prime}$ and the highest power raised to $y^{\prime}$ is one. Hence, its degree is one.

**9.** $y^{\prime \prime}+(y^{\prime})^{2}+2 y=0$

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**Solution**

$y^{\prime \prime}+(y^{\prime})^{2}+2 y=0$

The highest order derivative present in the differential equation is $y^{\prime \prime}$. Therefore, its order is two.

The given differential equation is a polynomial equation in $y^{\prime \prime}$ and $y^{\prime}$ and the highest power raised to $y^{\prime \prime}$ is one.

Hence, its degree is one.

**10.** $y^{\prime \prime}+2 y^{\prime}+\sin y=0$

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**Solution**

$y^{\prime \prime}+2 y^{\prime}+\sin y=0$

The highest order derivative present in the differential equation is $y^{\prime \prime}$. Therefore, its order is two.

This is a polynomial equation in $y^{\prime \prime}$ and $y^{\prime}$ and the highest power raised to $y^{\prime \prime}$ is one. Hence, its degree is one.

**11.** The degree of the differential equation

$\quad\quad$ $ (\frac{d^{2} y}{d x^{2}})^{3}+(\frac{d y}{d x})^{2}+\sin (\frac{d y}{d x})+1=0 \text{ is } $

$\quad\quad$(A) 3

$\quad\quad$(B) 2

$\quad\quad$(C) 1

$\quad\quad$(D) not defined

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**Solution**

$(\frac{d^{2} y}{d x^{2}})^{3}+(\frac{d y}{d x})^{2}+\sin (\frac{d y}{d x})+1=0$

The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.

Hence, the correct answer is D.

**12.** The order of the differential equation

$\quad\quad$ $ 2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0 \text{ is } $

$\quad\quad$(A) 2

$\quad\quad$(B) 1

$\quad\quad$(C) 0

$\quad\quad$(D) not defined

## Show Answer

**Solution**

$2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$

The highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, its order is two.

Hence, the correct answer is A.