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.
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 $latex y = vx$ to reduce the equation to a separable form, which can then be integrated to find the solution.
Differential Equations Class 12 Maths Notes PDF
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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:$latex \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 $latex y = vx$, where $latex v$ is a function of $latex x$. Substituting $latex y = vx$ into the differential equation gives: $latex \frac{dy}{dx} = v + x\frac{dv}{dx} $
Substituting this back into the original equation allows it to be rewritten in terms of $latex v$ and $latex x$: $latex v + x\frac{dv}{dx} = f(v) $
This equation can then be rearranged to separate the variables $latex v$ and $latex x$, leading to: $latex \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 $latex M(x, y)$ and $latex N(x, y)$ in $latex M(x, y)dx + N(x, y)dy = 0$ are homogeneous functions of the same degree.
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