Class 12 Differential Equations 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
An exact equation is a type of first-order differential equation that can be written in the form:$latex M(x, y) , dx + N(x, y) , dy = 0$The equation is said to be exact if there exists a function $latex f(x, y)$ such that:$latex M , dx + N , dy = d f(x, y)$.
In other words, $latex M(x, y)$ and $latex N(x, y)$ are the partial derivatives of some function $latex f(x, y)$ with respect to $latex y$ and $latex x$ respectively:$latex M(x, y) = \frac{\partial f(x, y)}{\partial y}$ and $latex N(x, y) = \frac{\partial f(x, y)}{\partial x}$.
If such a function $latex f(x, y)$ exists, the solution to the differential equation can be found by integrating $latex M$ and $latex N$ appropriately, leading to an expression for $latex f(x, y)$ equal to a constant..
Class 12 Differential Equations Notes PDF
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Exact Equations
A differential equation written in the form $latex M(x, y) , dx + N(x, y) , dy = 0$, where $latex M $ and $latex N $ are functions of $latex x $ or $latex y $ or both, is said to be exact if there exists a function $latex f(x, y) $ such that $latex M , dx + N , dy = d f(x, y) $,i.e., when $latex M , dx + N , dy $ is an exact or a perfect differential.
i.e., when $latex M , dx + N , dy $ is an exact or a perfect differential.The differential equation $latex y , dx + x , dy = 0 $ is exact, since $latex y , dx + x , dy = d(xy) = 0 $which gives,
latex xy = c $where $latex c $ is an arbitrary constant. But, the differential equation $latex x , dy – y , dx = 0 $is not exact as it stands.It, however, becomes exact if we multiply both sides of it by $latex \frac{1}{x^2} $, since, $latex \frac{1}{x^2} , (x , dy – y , dx) = 0 $ becomes $latex d \left(\frac{y}{x}\right) = 0 $.On integration, we have $latex \frac{y}{x} = c \quad \text{or} \quad y = cx $, where $latex c $ is an arbitrary constant as its solution.
An expression or factor such as $latex \frac{1}{x^2} $ is called an integrating factor (I.F.). Integrating factors may be found in several ways. But we shall focus our interest mainly on those cases in which $latex I.F. $ can be found by simple observations or inspection. One should not forget that an equation may be exact by simply regrouping various terms of the equation.
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