A first order first degree homogeneous differential equationa)Can always be reduced to an exact formb)Can never be reduced to an exact formc)May or may not. be reduced to an exact formd)Can only be reduced to an exact form if the given differential equation is linearCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

First Order First Degree Homogeneous Differential Equation

A first order first degree homogeneous differential equation is of the form:

dy/dx = f(x, y)

where f(x, y) is a homogeneous function of degree 1.

Reducing to an Exact Form

To determine whether a first order first degree homogeneous differential equation can be reduced to an exact form, we need to consider the concept of exactness.

A differential equation is said to be exact if it can be written in the form:

M(x, y) dx + N(x, y) dy = 0

where M(x, y) and N(x, y) are functions of x and y.

For a first order differential equation to be exact, it must satisfy the condition:

∂M/∂y = ∂N/∂x

In the case of a first order first degree homogeneous differential equation, the condition for exactness is automatically satisfied.

Explanation

A homogeneous differential equation of degree 1 can be written in the form:

dy/dx = f(x, y)

where f(x, y) is a homogeneous function of degree 1. Since f(x, y) is a homogeneous function, we can write it as:

f(x, y) = g(y/x)

Now, substituting y = vx into the differential equation, we get:

dy/dx = g(v)

Differentiating both sides with respect to x, we have:

d^2y/dx^2 = dg(v)/dv * dv/dx

Since the original differential equation is of first order, the second derivative term can be eliminated. Therefore, we have:

g(v) = c

where c is a constant.

Now substituting y = vx back into the equation, we get:

f(x, y) = g(y/x) = g(v) = c

Hence, the original first order first degree homogeneous differential equation can be reduced to the exact form:

dy/dx = c

Conclusion

A first order first degree homogeneous differential equation can always be reduced to an exact form. This is because the condition for exactness is automatically satisfied for such equations. Therefore, the correct answer is option A - 'Can always be reduced to an exact form.'

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams