Differential Equations Class 12 Maths Notes | Exercise - 15.3

Differential Equations Class 12 Maths Notes has been updated according to the latest syllabus of 2080. It means the notes of differential equations chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.

Contents

Chapter – 15
Differential Equations

In Class 12 Mathematics, homogeneous differential equations are a significant topic under differential equations. A differential equation is said to be homogeneous if it can be written in the form where each term is a homogeneous function of the same degree.

The method of solving these equations involves substituting $y = vx$ to reduce the equation to a separable form, which can then be integrated to find the solution.

Differential Equations Class 12 Maths Notes PDF

This PDF will provide the solutions of every question from the 3rd exercise of class 12 differential equation chapter. If you want the notes of other exercises then you can choose the exercise from the button given above.

Please repost this PDF on any website or social platform without permission.

Homogeneous Differential Equations

A differential equation of the first order and first degree is said to be homogeneous if it can be written in the form: $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$ .

Solving the Equation

To solve a homogeneous differential equation, the typical method is to make the substitution $y = vx$ , where $v$ is a function of $x$ . Substituting $y = vx$ into the differential equation gives: $\frac{dy}{dx} = v + x\frac{dv}{dx}$

Substituting this back into the original equation allows it to be rewritten in terms of $v$ and $x$ : $v + x\frac{dv}{dx} = f(v)$

This equation can then be rearranged to separate the variables $v$ and $x$ , leading to: $\frac{dv}{f(v) - v} = \frac{dx}{x}$ .

Finally, integrating both sides gives the solution to the homogeneous differential equation.

Test for Homogeneity

Checking if the function $M(x, y)$ and $N(x, y)$ in $M(x, y)dx + N(x, y)dy = 0$ are homogeneous functions of the same degree.

If the solutions PDF of differential equations chapter was helpful to you then feel free to leave your comments sharing your thought and opinions. You can also join our telegram channel to remain connected with us. We keep on posting all the latest news and notes in our telegram group. So we’ll be happy if you be a part of it.