Exact Differential Equation: Overview, Questions, Preparation

Differential Equations 2021 ( Differential Equations )

Rachit Kumar Saxena

Rachit Kumar SaxenaManager-Editorial

Updated on Jun 29, 2021 02:06 IST

What is Exact Differential Equation?

Differential equations are equations that tell one or more of their functions and derivatives. In applications, functions generally represent the number of physics, the derivative represents their level of change, and differential equations define a relationship between the two. The equation P (x,y) dx + Q (x,y) dy=0 is an exact differential equation if there exists a capacity f of two variables x and y having consistent halfway derivatives with the end goal that the exact differential equation definition is isolated as follows. 

ux(x, y) = p(x, y) and uy (x, y) = Q(x, y); 

Thus, the overall arrangement of the equation is u(x, y) = C. 

Where "C" is a discretionary steady.

Exactness

Expect the capacities P(x, y) and Q(x, y) to have the nonstop fractional derivatives in a specific area D, and the differential equation is exact if and just if it fulfils the condition. 

∂Q/∂x=∂P/∂y

Integrating Factor

If the differential equation P (x, y) dx + Q (x, y) by = 0 is not exact, it is conceivable to make it exact by increasing utilising a pertinent factor u(x, y), which is known as coordinating element for the given differential equation. 

Think about a model, 

2ydx + x dy = 0 

Presently, check whether the given differential equation is exact utilising testing for exactness. 

The given differential equation is not exact. 

To change it over into the exact differential equation, increase by the incorporating factor u(x,y)= x, the differential equation becomes, 

2 xy dx + x2 dy = 0 

The above resultant equation is the exact differential equation because the left half of the equation is an absolute differential of x2y. 

It is hard to track down the incorporating factor. In any case, there are two classes of differential equations whose incorporating components might be found without any problem. Those equations have the coordinating variable having the elements of either x alone or y alone. 

Weightage of Exact Differential Equation

In class 12: The chapter Differential Equations deals with the various methods and theorems to solve the equations and the question regarding them. It has a weightage of 35 Marks.

Illustrated Examples on Exact Differential Equation

1. Determine the order and degree of the differential equation 2x(d^4 y)/〖dy〗^4+5x^2 (dy/dx)^3-xy=0.
Solution.

Fourth-order, since the highest derivative in the equation, is the 4th derivative. First-order, since the exponent or power of the 4th derivative, is 1.

2. Classify the following differential equation: exdy dx + 3y = x2y
Solution.

The equation can be written as 1 y dy dx = e−x(x2 − 3), which shows that it is separable. It can also be written as dy dx + e−x(3 − x2)y = 0, which is linear.

3. Suppose y is a function of x. Which of the following is d(x3y) dx?
Solution.

3x2y + x3dy dx, by using the production rule.

FAQs on Exact Differential Equation

Q: What are the types of differential equations?

A: The different types of differential equations may include Ordinary Differential Equations, Partial Differential Equations, Linear Differential Equations, Nonlinear differential equations, Homogeneous Differential Equations, Nonhomogeneous Differential Equations.  

Q: Can we separate all exact equations?

A: The equation that can be separated is always right, because according to their definition of form: M (Y) Y + N (T) = 0, and then if a (y), b (t) is an antiderivative of M and N (RESP.). This is the same as: (a (y) + b (t)) = 0, so φ (t, y) = a (y) + b (t) is the amount preserved.

Q: What are the applications of differential equations?

A: The investigation of differential equations is a wide field in unadulterated and applied arithmetic, physical science, and designing. Numerous principal laws of material science and science can be detailed as differential equations. In science and financial matters, differential equations are utilised to demonstrate the conduct of complex frameworks.

Q: Can we say that dy/dx equal to y?

 A: Yes, we can say that because d/dx (y) is an equivalent to dy/dx . df/dx is an expression that means "the derivative of f, with respect to x".

Q: Define differential equation of first order.

A:  A first request differential equation is an equation of the structure F(t,y,˙y)=0. An answer of a first request differential equation is a capacity f(t) that makes F(t,f(t),f′(t))=0 for each estimation of t. Here, F is an element of three variables: t, y, and ˙y.
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Differential Equations Exam

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