RD Sharma Class 12 Ex 22.1 Solutions Chapter 22 Differential Equations

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TextbookNCERT
ClassClass 12th
SubjectMaths
Chapter22
Exercise22.1
CategoryRD Sharma Solutions

RD Sharma Class 12 Ex 22.1 Solutions Chapter 22 Differential Equations

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,

\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, 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 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

Order 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.

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

Order 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 + e= 0 

Solution:  

We have,

 dy/dx + e= 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”)+ (y’)3 + siny = 0

Solution:

We have,

(y”)+ (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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