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
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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.
