The differential equation 2y dx – (3y – 2x) dy = 0 is

This question was previously asked in
BPSC Asstt. Prof. ME Held on Nov 2015 (Advt. 22/2014)
View all BPSC Assistant Professor Papers >
  1. exact and homogenous but not linear
  2. exact, homogenous and linear
  3. exact and linear but not homogenous
  4. homogenous and linear but not exact

Answer (Detailed Solution Below)

Option 2 : exact, homogenous and linear

Detailed Solution

Download Solution PDF

Concept:

Homogenous equation: If the degree of all the terms in the equation is the same then the equation is termed as a homogeneous equation.

​Exact equation: The necessary and sufficient condition of the differential equation M dx + N dy = 0 to be exact is:

\(\frac{{\partial M}}{{\partial y}} = \frac{{\partial N}}{{\partial x}}\)

Linear equation: A differential equation is said to be linear if the dependent variable and its differential coefficient only in the degree and not multiplied together.

The standard form of a linear equation of the first order, commonly known as Leibnitz's linear equation is:

 \(\frac{{dy}}{{dx}}+Py=Q\) 

where, P, Q is a function of x.

or, \(\frac{{dx}}{{dy}}+Px=Q\)

where, P, Q is a function of x.

Condition 1:

2y dx + (2x - 3y) dy = 0   ---.(1)

(It is Homogeneous) 

Condition 2:

Equation (1) can be written as  ​\(\frac{{dy}}{{dx}}=\frac{{2y}}{{2x\;-\;3y}}\) .

It is not a linear form.

or \(\frac{{dx}}{{dy}}=\frac{{2x-3y}}{{2y}}\)

\(\frac{{dx}}{{dy}}+\frac{{x}}{{y}}=\frac{{3}}{{2}}\)

It is in linear form

Condition 3:

M dx + N dy = 0

2y dx – (3y – 2x) dy = 0

hence, M = 2y and N = 2x - 3y

\(\frac{{\partial M}}{{\partial y}} =\frac{{\partial (2y)}}{{\partial y}}= 2\) and \(\frac{{\partial N}}{{\partial x}}= \frac{{\partial (2x+3y)}}{{\partial x}}=2\)

As \(\frac{{\partial M}}{{\partial y}}=\frac{{\partial N}}{{\partial y}}\) 

so, it is an exact equation.

Latest BPSC Assistant Professor Updates

Last updated on May 9, 2025

-> The BPSC Assistant Professor last date to apply online has been extended to 15th May 2025 (Advt. No. 04/2025 to 28/2025).

-> The BPSC Assistant Professor Notification 2025 has been released for 1711 vacancies under Speciality (Advt. No.04/2025 to 28/2025).

-> The recruitment is ongoing for 220 vacancies (Advt. No. 01/2024 to 17/2024).

-> The salary under BPSC Assistant Professor Recruitment is approximately Rs 15600-39100 as per Pay Matrix Level 11. 

-> The selection is based on the evaluation of academic qualifications &  work experience and interview.

-> Prepare for the exam using the BPSC Assistant Professor Previous Year Papers. For mock tests attempt the BPSC Assistant Professor Test Series.

More Differential Equations Questions

Get Free Access Now
Hot Links: teen patti master download teen patti 3a teen patti baaz