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

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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.

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Last Updated : 12 Dec, 2021

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

Determine the order and degree of the following differential equation. State also whether it is linear or non-linear(Question 1-13)

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

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

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

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

Question 5. $\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+xy=0$

Question 6. $3\sqrt{\frac{d^2y}{dx^2}}= \sqrt\frac{dy}{dx}$

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

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

Question 9: $y\frac{d^2x}{dy^2}=y^2+1$

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

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

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

Question 13: $(xy^2+x)dx+(y-x^2y)dy=0$

Data Structures and Algorithms - Self Paced

CBSE Class 12 Computer Science

GATE CS & IT 2024

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

Determine the order and degree of the following differential equation. State also whether it is linear or non-linear(Question 1-13)

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8:

Question 9:

Question 10:

Question 11:

Question 12:

Question 13:

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Data Structures and Algorithms - Self Paced

CBSE Class 12 Computer Science

GATE CS & IT 2024

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

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

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

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

Question 5. $\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+xy=0$

Question 6. $3\sqrt{\frac{d^2y}{dx^2}}= \sqrt\frac{dy}{dx}$

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

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

Question 9: $y\frac{d^2x}{dy^2}=y^2+1$

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

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

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

Question 13: $(xy^2+x)dx+(y-x^2y)dy=0$