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Class 12 RD Sharma Solutions- Chapter 22 Differential Equations – Exercise 22.1 | Set 1

  • 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 1-13)

Question 1. \frac{d^3x}{dt^3}+\frac{d^2x}{dt^2}+(\frac{dx}{dt})^2=e^t

Solution:

We have,

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\frac{d^3x}{dt^3}+\frac{d^2x}{dt^2}+(\frac{dx}{dt})^2=e^t



Order of function:

The Highest order of derivative of function is 3 i.e.,( \frac{d^3x}{dt^3})

So, the order of derivative is equal to 3.

Degree of function:

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

So, degree of function is 1.

Linear or Non-linear:

The given equation is non-linear.



Question 2. \frac{d^2y}{dx^2}+4y=0

Solution:

We have,

\frac{d^2y}{dx^2}+4y=0

Order of function:

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

So, Degree of the function is equal to 1.

Linear or Non-linear:



The given equation is linear.

Question 3. (\frac{dy}{dx})^2+\frac{1}{(\frac{dy}{dx})}=2

Solution:

We have,

 (\frac{dy}{dx})^2+\frac{1}{(\frac{dy}{dx})}=2

(\frac{dy}{dx})^3+1=2(\frac{dy}{dx})

 (\frac{dy}{dx})^3-2(\frac{dy}{dx})+1=0

Order of function:

As the highest order of derivative of function is 1 (i.e., \frac{dy}{dx})

So, 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.

Question 4. \sqrt{[1+(\frac{dy}{dx})^2]} =(c\frac{d^2y}{dx^2})^\frac{1}{3}

Solution:

We have,

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

On squaring both side, we get

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

On cubing both side, we get



1+3(\frac{dy}{dx})^2+3(\frac{dy}{dx})^4+(\frac{dy}{dx})^6=c^2(\frac{d^2y}{dx^2})^2

c^2(\frac{d^2y}{dx^2})^2-1-3(\frac{dy}{dx})^2-3(\frac{dy}{dx})^4-(\frac{dy}{dx})^6=0

Order of function:

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

So, Oder 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 5. \frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+xy=0

Solution:

We have,

\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+xy=0

Order of function:

As the highest order of derivative of function is 2 

So, Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of 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 6. 3\sqrt{\frac{d^2y}{dx^2}}= \sqrt\frac{dy}{dx}

Solution:  

We have,

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

On cubing both side, we get

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

On squaring both side, we get

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

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 7. \frac{d^4y}{dx^4}=[c+(\frac{dx}{dy})^2]^\frac{3}{2}

Solution:

We have,

\frac{d^4y}{dx^3}=[c+(\frac{dy}{dx})^2]^\frac{3}{2}

On squaring both side, we get

(\frac{d^4y}{dx^4})^2=[c+(\frac{dy}{dx})^2]^3

(\frac{d^4y}{dx^4})^2=[c^3+(\frac{dy}{dx})^6+3c(\frac{dy}{dx})^2+3c^2(\frac{dy}{dx})]

(\frac{d^4y}{dx^4})^2-(\frac{dy}{dx})^6-3c(\frac{dy}{dx})^2-3c^2(\frac{dy}{dx})-c^3=0

Oder of function:

The highest order of derivative of function is 4 (i.e., \frac{d^4y}{dx^4})

So, the order of the derivative is equal to 4.

Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of \frac{d^4y}{dx^4} is 2)

So, the degree of function is 2.

Linear or Non-linear:



The given equation is non-linear.

Question 8: x+\frac{dy}{dx}=\sqrt{1+(\frac{dy}{dx})^2}

Solution:

We have,

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

On squaring both side, we have

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

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

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

\frac{dy}{dx}+\frac{x}{2}-\frac{1}{2x}=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 1.

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

Linear or Non-linear:

The given equation is linear.

Question 9: y\frac{d^2x}{dy^2}=y^2+1

Solution:

We have,

y\frac{d^2x}{dy^2}=y^2+1



\frac{d^2x}{dy^2}-y-\frac{1}{y}=0

Order of function:

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

So, order of derivative 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^2x}{dy^2}  is 1)

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

Linear or Non-linear:

The given equation is linear.

Question 10: s^2\frac{d^2t}{ds^2}+st\frac{dt}{ds}=s

Solution:

We have,

s^2\frac{d^2t}{ds^2}+st\frac{dt}{ds}=s

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^2t}{ds^2} is 1)

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

Linear or Non-linear:

The given equation is non-linear.

Question 11: x^2(\frac{d^2y}{dx^2})^3+y(\frac{dy}{dx})^4+y^4=0

Solution:

We have,

x^2(\frac{d^2y}{dx^2})^3+y(\frac{dy}{dx})^4+y^4=0

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 3. (i.e., power of \frac{d^2y}{dx^2}   is 3)

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

Linear or Non-linear:



The given equation is non-linear.

Question 12: \frac{d^3y}{dx^3}+(\frac{d^2y}{dx^2})^3+(\frac{dy}{dx})+4y=siny

Solution:

We have,

\frac{d^3y}{dx^3}+(\frac{d^2y}{dx^2})^3+(\frac{dy}{dx})+4y=siny

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 non-linear.

Question 13: (xy^2+x)dx+(y-x^2y)dy=0

Solution:

We have,

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

(xy^2+x)dx+(y-x^2y)dy=0

(y-x^2y)\frac{dy}{dx}+xy^2+x=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 Order of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.




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