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Question 511 Mark

Find the sum of the order and degree of the differential equation $\text{y}=\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)^{3}+\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}.$

Answer

The order is 2 as the highest derivative is 2.
The degree is 1 as the highest derivative is of order 1.
Hence, the sum of the order and dergree of the differential equation $\text{y}=\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)^{3}+\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$ is 2 + 1 = 3.

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Question 521 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\text{x}^2\log\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)$

Answer

$\frac{\text{dy}}{\text{dx}}+\text{e}^\text{x}=0$
The highest order differential coefficient is $\frac{\text{dy}}{\text{dx}}$ and its power is 1.
So, it is a non linear differential equation of order 1 and degree 1.

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Question 531 Mark

Determine order and degree (if defined) of differential equations given in Exercise.
$y' + y = e^x$

Answer

The given differential equation is
$y' + y = e^x$
The highest order derivative present in the given differential equation is y' and index of its highest power is 1.
$\therefore$ the given differential equation is of order 1 and degree 1.

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Question 541 Mark

Write the order of the differential equation of the famliy of circles of radius r.

Answer

Given, the equation of famliy of circle,
$(\text{x}-\text{a}^{2})+(\text{y}-\text{b})^{2}=\text{r}^{2}\ ...(\text{i})$
Since, given equation have two arbiteary constant, so we differential the above equation two times wrt.x.
Differential equation (i) wrt.x. we get
$2(\text{x}-\text{a})+2(\text{y}-\text{b})\frac{\text{}dy}{\text{dx}}=\text{r}^{2}=0$
$\Rightarrow (\text{x}-\text{a})+(\text{y}-\text{b})\frac{\text{dy}}{\text{dx}}=0\ ...(\text{ii})$
Differential equation (ii) wrt.x. we get,
$1+(\text{y}-\text{b})\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\big(\frac{\text{dy}}{\text{dx}}\big)=0$
$\Rightarrow(\text{y}-\text{b})=\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\ ...(\text{iii})$
Substitite value of (y - b) in equation (ii), we have,
$(\text{x}-\text{a})+\Bigg[\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\Bigg]\frac{\text{dy}}{\text{dx}}=0$
$(\text{x}+\text{a})+\frac{\Big[ 1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}\Big]\frac{\text{dy}}{\text{dx}}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\ ...(\text{iv})$
Substitite value of (x - a) (y - b) in equation (ii), we have,
$\Bigg[\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\Bigg]\frac{\text{dy}}{\text{dx}}+\Bigg[\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\Bigg]\frac{\text{dy}}{\text{dx}} =\text{r}^{2}$
$\Rightarrow \Big[1+\big(\frac{\text{dy}}{\text{dx}}\big)^{2}\Big] ^{3}=\text{r}^{2}\Big(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big)^{2}$
The order of the differential equation of the famliy of circle of redius r is 2.

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Question 551 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}+\text{y}\sin\text{y}=0$

Answer

$\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}+\text{y}\sin\text{y}=0$
The highest order differential coefficient is $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ and its power is 1.

So, it is non linear differential equation with order 3 and degree 1.

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Question 561 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{dy}}{\text{dx}}+\text{e}^\text{y}=0$

Answer

$\frac{\text{dy}}{\text{dx}}+\text{e}^\text{y}=0$
In this differential equation, the order of the highest order derivative is 1 and its power is 1. So, it is a differential equation of order 1 and degree 1.
It is a non-linear differential equation, as the exponent of the dependent variable is not equal to 1(as per expansion series series of $e^y$.

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Question 571 Mark

Write the degree of the following differrntial equation $ \Big(\frac{\text{dy}}{\text{dx}}\Big)^{4}+3\text{x}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=0.$

Answer

$ \Big(\frac{\text{dy}}{\text{dx}}\Big)^{4}+3\text{x}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=0$
Here, we see that the highest order derivative is $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$ and its power is 1.
Therefore, the given differential equation is of first degree.

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Question 581 Mark

Write the differential equation representing the famliy of straight line y = Cx + 5, where C is an arbitrary constant.

Answer

We have,
$\text{y}=\text{Cx}+5\ ...(\text{i})$
$\Rightarrow \frac{\text{dy}}{\text{dx}}=\text{C}$
Substituting the value of C in (i),
$\text{y}=\frac{\text{dy}}{\text{dx}}\times\text{x}+5$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}+5=0$
Hence, $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}+5=0$ is the differential equation tha famliy of lines $\text{y}=\text{Cx}+5$ When C is an arbita.

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Question 591 Mark

Write the degree of the differrntial equation $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}=\text{x}^{2}\log\Big(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big).$

Answer

The given differential equation is not a polnomial equation in derivaties.
Hence, the degree for this differential equation is not defind.

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Question 601 Mark

Write the differrntial equation representing famliy of curve y = mx, where m is arbitrary constant.

Answer

We have,
$\text{y}=\text{mx}\ ...(\text{i})$
Differentiating with respect to x,
$\Rightarrow \frac{\text{dy}}{\text{dx}}=\text{m}$
Substituting the value of $\frac{\text{dy}}{\text{dx}}=\text{m}$ in equation (i),
$\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}$
Hence, $\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}$ is the required differential equation.

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Question 611 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$

Answer

$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$
The order of a differential equation is the order of the highest order derivative appearing in the equation. The degree of a differential equation is the degree of the highest order derivative. Consider the given differential equation

$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$

In the above equation, the order of the highest order derivative is 1. So the differential equation is of order 1. In the above differential equation, the power of the highest order derivative is 3. Hence, it is a differential equation of degree 3. Since the degree of the above differential equation is 3, more than one, it is a non-linear differential equation.

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Question 621 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\text{x}\sin\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)$

Answer

$\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\text{x}\sin\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)$
In this differential equation, the order of the highest order derivative is 2.

Clearly, the R.H.S. of the differential equation cannot be expressed as a polynomial in $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ So, its degree is not defined.

The order of the differential equation is 2 and its degree is not defined.

It is a non-linear differential equation, as one of its differential co-efficients, that is, $\Big(\frac{\text{dy}}{\text{dx}}\Big)$ has exponent 2, which is more than 1.

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Question 631 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{x}^2\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)^3+\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)^4+\text{y}^4=0$

Answer

$\text{x}^2\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)^3+\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)^4+\text{y}^4=0$
In this differential equation, the order of the highest order derivative is 2 and its power is 3. So, it is a differential equation of order 2 and degree 3.

It is a non linear differential equation, as its degree is more than 1.

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Maths 1 Marks Question