Differential equation questions and answers will help students quickly grasp the fundamentals of the concept. Students can use these questions to quickly summarize the concepts and practice answering them in order to increase their understanding. You can also acquire comprehensive answers to each question to verify your answers. To find out more about differential equations, click here.
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What are Differential Equations? In calculus, an equation which involves the derivatives of the dependent variable with reference to the independent variable is referred to as a differential equation. The differential equation assists us in presenting a relation between the changing quantity with regard to the change in the other quantity. The derivative represents the rate of change. Be a function with the form: y=f(x), where x is an independent variable, and y is the dependent variable. Also, read: Derivatives. |
Differential Equation Questions with Solutions
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First Order Differential Equation: The first-order differential equation includes all linear equations that take the form of derivatives. It only has the first derivative, which is expressed as dy/dx or y’ or f(x, y), where x and y are the two variables. Second-Order Differential Equation: The second-order differential equation is the equation that has second-order derivatives. The second-order differential equation is expressed as d/dx(dy/dx) or f”(x) or d2y/dx2 or y”. Also, read: Order of Differential Equations. |
1. What is the order of the differential equation dy/dx + y = 3y2.
Solution:
Given: dy/dx + y = 3y2
The order of the differential equation is 1st order.
In the given differential equation, dy/dx represents the first-order differential equation.
Hence, the order of the differential equation dy/dx + y = 3y2 is 1.
2. Find the order of the differential equation y”’ + y”y’ = 3x2.
Solution:
Given differential equation is y”’ + y”y’ = 3x2.
Here, the highest order of the given differential equation is 3.
Because, in the given differential equation, y”’ = d3y/dx3, which denotes the third-order differential equation.
Therefore, the order of the differential equation y”’ + y”y’ = 3x2 is 3.
3. Find the order and degree of the differential equation y’+5y = 0.
Solution:
Given: y’ + 5y = 0
The highest order of the derivative is 1.
Therefore, order = 1.
The degree of the differential equation = power of y’
Hence, degree = 1.
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Ordinary Differential Equation: In Calculus, the Ordinary Differential Equation (ODE) is defined as an equation which has only one independent variable and it can have one or more derivatives with regard to the variable. An ODE can be either homogeneous or non-homogeneous. Generally, the ODEs can be expressed using the relation which contains one independent variable (let’s say x) and the real dependent variable (let’s say y), and a few of its derivatives (say, y, y’, y”,…yn) with respect to the independent variable x. Partial Differential Equation: The partial differential equation, also known as PDE, is defined as an equation which involves one or many unknown multi variables with partial derivatives of one or many functions. Generally, in PDEs, the partial derivatives are expressed as ux = ∂u/∂x. |
4. Differentiate the function 10x2 with respect to x.
Solution:
Let the function be “y”.
Hence, y = 10x2
Now, differentiate the function y = 10x2 with respect to x, and we get
dy/dx = 2(10)x [As (d/dx)xn = nxn-1]
Hence, dy/dx = y’ = 20x.
5. Find the derivative of sin(3x+5)
Solution:
Let y = sin (3x+5)
As we know, the (d/dx) sin x = cos x.
Hence, dy/dx = cos (3x+5). (3) [Since, differentiation of 3x is 3]
Thus, dy/dx = 3 cos (3x + 5)
Hence, the derivative of sin (3x + 5) is 3 cos (3x+5).
6. Show that x = elog x is true for all the real x?
Solution:
Given equation: x = elog x.
If x = 0, then the equation becomes,
0 = elog 0
But log 0 is not defined, and hence the equation x = elog x is not defined for x = 0.
If x < 0, log x is not defined for negative numbers and hence the equation x = elog x is not defined for x < 0.
If x > 0, log x is defined for x > 0.
Hence, the equation x = e log x is defined for x > 0.
This shows that the equation x = elog x is true for all positive values of x.
7. What is the number of arbitrary constants in the general solution of the given differential equation, if its order is 4?
Solution:
For example, the differential equation of fourth order is y”” = 2.
Hence, its general solution becomes
y = (x4/24) + (Ax3/6) + (Bx2/2) + Cx + D
Thus, the number of constants is 4.
Therefore, the number of arbitrary constants is 4.
In other words, the order of differential equations is equal to the number of arbitrary constants.
Since the order of the given differential equation is 4, the number of the arbitrary constants should be 4.
8. Find the number of arbitrary constants in the particular solution of the given differential equation if its order is 3.
Solution:
As we know, in a particular solution, there should be any arbitrary constants.
Therefore, the number of arbitrary constants in the particular solution of the differential equation of third order is 0.
9. Find the integrating factor of the differential equation x(dy/dx) – y = 2x2.
Solution:
Given the differential equation: x(dy/dx) – y = 2x2
Now, divide both sides of the equation by x, we get
(x/x)(dy/dx) – (y/x) = (2x2/x)
(dy/dx) – (y/x) = 2x …(1)
Hence, the differential equation is of the form:
(dy/dx) + Py = Q …(2)
Comparing the equations (1) and (2), we get
P = -1/x, and Q = 2x.
We know that the Integrating factor, IF = e∫Pdx
Now, substitute the obtained values in the above formula, and we get
IF = e∫ (-1/x)dx
IF = e– log x
As, x log y = log yx,
Hence, IF = x-1 [Since elog x = x]
IF = 1/x [As, x-1 = 1/x]
Therefore, the integrating factor (IF) of the differential equation x(dy/dx) – y = 2x2 is 1/x.
10. Find the second order derivative of x20.
Solution:
Assume that y = x20.
Now, differentiate the function with respect to x, we get
dy/dx = (d/dx)(x20) = 20x19. [Since (d/dx)xn = nxn-1]
Again differentiate the above function to find the second derivative.
(d2y/dx2) = (d/dx)(20x19)
(d2y/dx2) = (20).(19) x18
(d2y/dx2 ) = 380x18.
Hence, the second order derivative of x20 is 380x18.
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Practice Questions
Solve the following differential equation questions.
- What is the second order derivative of log x.
- Find the degree of the differential equation y = x(dy/dx)2 + (dy/dx).
- What is the general solution of xy(dy/dx) – 1 = 0.
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