Determine the order and degree of the following differential equation. State also whether it is linear or non-linear(Questions 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.
Practice Questions
1. Verify that y = e^(x^2) is a solution of the differential equation dy/dx = 2xy.
2. Find the order and degree of the differential equation: (d^2y/dx^2 + 1)^3 = 0
3. Form the differential equation of the family of curves y = ax^2 + bx + c, where a, b, and c are arbitrary constants.
4. Solve the differential equation: dy/dx + y tan x = sec x
5. Find the general solution of the differential equation: dy/dx = x/y
6. Verify that y = sin x + cos x is a solution of the differential equation d^2y/dx^2 + y = 0
7. Solve the differential equation: (1 + x^2)dy/dx = 1 - y^2
8. Find the particular solution of dy/dx = 2x, given that y = 1 when x = 0.
9. Form the differential equation of all circles with center at the origin.
10. Solve the differential equation: dy/dx = (y^2 - 1)/(2xy)
Summary
Chapter 22 of RD Sharma's Class 12 mathematics textbook focuses on Differential Equations.
Questions
- Definition of a differential equation
- Order and degree of a differential equation
- Formation of differential equations by eliminating arbitrary constants
- Solutions of first-order differential equations
- Methods for solving separable variables
- Homogeneous differential equations
- Linear differential equations
- Exact differential equations
This chapter provides a foundation for understanding and solving various types of differential equations, which are crucial in many areas of mathematics, physics, and engineering.
