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Class 12 RD Sharma Solutions – Chapter 22 Differential Equations – Exercise 22.1 | Set 2
  • Last Updated : 21 Feb, 2021

Determine the order and degree of the following differential equation. State also whether it is linear or non-linear(Question 14-26)

Question 14. \sqrt{1-y^2}dx+\sqrt{1-x^2}dy=0

Solution:  

We have,

\sqrt{1-y^2}dx+\sqrt{1-x^2}dy=0

\frac{dy}{dx}+ \sqrt{\frac{1-y^2}{1-x^2}}=0

Oder of function:



As the highest order of derivative of function is 1 (i.e., dy/dx)

So, the order of the derivative is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of dy/dx is 1)

So, the degree of function is 1.

Linear or Non-linear:

The given equation is non-linear.

Question 15. \frac{d^2y}{dx^2}=(\frac{dy}{dx})^\frac{2}{3}

Solution:



We have,

\frac{d^2y}{dx^2}=(\frac{dy}{dx})^\frac{2}{3}

On cubing both side, we have

(\frac{d^2y}{dx^2})^3=(\frac{dy}{dx})^2

Order of function:

The Highest order of derivative of function is 2. (i.e., \frac{d^2y}{dx^2})

So, the order of the derivative is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 3 (i.e., power of \frac{d^2y}{dx^2} is 3)

So, the degree of function is 3.

Linear or Non-linear:

The given equation is non-linear.

Question 16. 2\frac{d^2y}{dx^2}+3*\sqrt{1-(\frac{dy}{dx})^2-y}=0

Solution:

We have,

2\frac{d^2y}{dx^2}+3*\sqrt{1-(\frac{dy}{dx})^2-y}

2\frac{d^2y}{dx^2}=-3*\sqrt{1-(\frac{dy}{dx})^2-y}

Squaring both sides, we have

4(\frac{d^2y}{dx^2})^2=9({1-(\frac{dy}{dx})^2-y})

4(\frac{d^2y}{dx^2})^2+9(\frac{dy}{dx})^2+9y-9=0

Order of function:

As the highest order of derivative of function is 2. (i.e., \frac{d^2y}{dx^2})

So, the order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of \frac{d^2y}{dx^2} is 2)

So, the Degree of the function is equal to 2.

Linear or Non-linear:

The given equation is non-linear.

Question 17. 5\frac{d^2y}{dx^2}=[1+(\frac{dy}{dx})^2]^\frac{3}{2}

Solution:

We have,

5\frac{d^2y}{dx^2}=[1+(\frac{dy}{dx})^2]^\frac{3}{2}

One squaring both side, we have

[5\frac{d^2y}{dx^2}]^2=[1+(\frac{dy}{dx})^2]^3

25(\frac{d^2y}{dx^2})^2=[1+(\frac{dy}{dx})^6+3(\frac{dy}{dx})^2+3(\frac{dy}{dx})^6

Order of function:

As the highest order of derivative of the function is 2 (i.e., \frac{d^2y}{dx^2})

So, Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of \frac{d^2y}{dx^2} is 2)

So, the Degree of the function is equal to 2.

Linear or Non-linear:

The given equation is non-linear.

Question 18. y=x\frac{dy}{dx}+a\sqrt{1+(\frac{dy}{dx})^2}

Solution:

We have,

y=x\frac{dy}{dx}+a\sqrt{1+(\frac{dy}{dx})^2}

(y-x\frac{dy}{dx})=-a\sqrt{1+(\frac{dy}{dx})^2}

On squaring both sides, we get

(y-x\frac{dy}{dx})^2=a(1+(\frac{dy}{dx})^2

x^2(\frac{dy}{dx})^2+y^2-2xy\frac{dy}{dx}=a[1-(\frac{dy}{dx})^2]

x^2(\frac{dy}{dx})^2+y^2-2xy\frac{dy}{dx}-a+(\frac{dy}{dx})^2=0

Order of function:

As the highest order of derivative of the function is 1,

So, the Order of the function is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 2.(i.e., power of dy/dx is 2)

So, the Degree of the function is equal to 2.

Linear or Non-linear:

The given equation is non-linear.

Question 19. y=px+\sqrt{a^2p^2+b^2} , where p = dy/dx 

Solution:

We have

y=px+\sqrt{a^2p^2+b^2} , where p = dy/dx 

(y-px)^2={a^2p^2+b^2}

y^2-2pxy+p^2x^2=a^2p^2+b^2

(x^2-a^2)(\frac{dy}{dx})^2-2xy(\frac{dy}{dx})+(y^2-b^2)=0

Order of function:

As the highest order of derivative of function is 1

So, the Order of the function is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of dy/dx is 2)

So, the Degree of the function is equal to 2.

Linear or Non-linear:

The given equation is non-linear.

Question 20: dy/dx + ey = 0 

Solution:  

We have,

 dy/dx + ey = 0 

Order of function:

As the highest order of derivative of the function is 1

So, the Order of the function is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 1(i.e., power of dy/dx is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.

Question 21. (\frac{d^2y}{dx^2})^2+(\frac{dy}{dx})=xsin(\frac{d^2y}{dx^2})

Solution:  

We have,

(\frac{d^2y}{dx^2})^2+(\frac{dy}{dx})=xsin(\frac{d^2y}{dx^2})

Order of function:

As the highest order of derivative of the function is 2

So, the order of the derivative is equal to 2.

Degree of function:

sin(\frac{d^2y}{dx^2}) is not a polynomial function. So degree can not be defined. 

So, the degree of function is not defined.

Linear or Non-linear:

The given equation is non-linear.

Question 22. (y”)2 + (y’)3 + siny = 0

Solution:

We have,

(y”)2 + (y’)3 + siny = 0

Where

 y''=\frac{d^2y}{dx^2}

Order of function:

The highest order of derivative of the function is 2. (i.e., y”)

So, the order of the derivative is equal to 2.

Degree of function

As the power of the highest order derivative of the function is 2 (i.e., power of y” is 2)

So, the degree of function is 2.

Linear or Non-linear:

The given equation is non-linear.

Question 23.\frac{d^2y}{dx^2}+5x(\frac{dy}{dx})^2-6y=logx

Solution:

We have,

\frac{d^2y}{dx^2}+5x(\frac{dy}{dx})^2-6y=logx

Order of function:

As the highest order of derivative of the function is 2. 

So, the order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of \frac{d^2y}{dx^2} is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.

Question 24. \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+ysiny=0

Solution:

We have,

\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+ysiny=0

Order of function:

As the highest order of derivative of the function is 3 

So, the Order of the function is equal to 3.

Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of\frac{d^3y}{dx^3} is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is linear.

Question 25. \frac{d^2y}{dx^2}+3(\frac{dy}{dx})^2 =x^2log(\frac{d^2y}{dx^2})

Solution:

We have,

\frac{d^2y}{dx^2}+3(\frac{dy}{dx})^2 =x^2log(\frac{d^2y}{dx^2})

Order of function:

As the highest order of derivative of the function is 2.

So, the order of the function is equal to 2.

The degree of function:

log(\frac{d^2y}{dx^2})  is not a polynomial function. So degree can not be defined. 

So, the degree of function is not defined.

Linear or Non-linear:

The given equation is non-linear.

Question 26. (\frac{dy}{dx})^3-4(\frac{dy}{dx})^2+7y=sinx

Solution:

We have,

(\frac{dy}{dx})^3-4(\frac{dy}{dx})^2+7y=sinx

\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+ysiny=0

Order of function:

As the highest order of derivative of the function is 1

So, the Order of the function is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 3(i.e., power of dy/dx is 3)

So, the Degree of the function is equal to 3.

Linear or Non-linear:

The given equation is non-linear.

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