Differential equations is a scoring topic from JEE Main point of view as every year 1 question is certainly asked. Every candidate should take care of not letting go easy marks from this topic. To revise effectively read and revise from the Differential Equations Short Notes. You can also download Indefinite Integration notes PDF at end of the post.
1. Differential Equations
A differential equation is a just as a normal equation consists of variables and numeric constants. The only difference in between the normal equation and differential equation is that the former contains one variable and constants whereas, in the differential equation, it consists of independent variables, dependent variables, and numeric constants only.
NOTE: Numeric constant means numbers
Examples of the differential equation:
2. Order and degree of a differential equation
Order of differential equation is defined as the highest number of times the dependent variable is differentiated with respect to the independent variable.
The degree of a differential equation is defined as the highest power of the highest order differential variable in the equation.
For example: in the equation, the order is 3 and the degree is 1.
3. Formation of a differential equation from a given equation
Whenever an equation is given, choose the dependent variable and independent variable.
Then differentiate the equation with respect to the independent variable. Now try and eliminate the constants and variables other than the dependent and differential term.
NOTE: Differentiate the equation as many numbers of variable or constants in the equation
Example: (ax+y2) =0. Find differential form
On differentiating w.r.t x we get
Thus, substituting in parent equation we get
4. General format of First order differential form
(a). Variable Separable form
If the equation is given in the form
and such that
Then this technique is applicable
Solution: Now we can separate in the following format
Integrating both sides, we get
ey=ex + C
(b) Differential equations reducible to a separable variable type
If the given equation is in the form of
; Then apply this technique.
Put ax+by=t , then differentiating we get
Substitute value of t and in terms ofand solve the given problem.
Putting in the equation
Now integrate both sides to get
5. HOMOGENEOUS FORM
A function f(x) is said to be in homogeneous form if the independent variable can be expressed in terms of the dependent variable and vice-versa.
In this type consider y=vx and hence
Substitute these values in the equation and obtain the result.
6. Exact Differential form
Certain differential equation need not be rearranged or solved using the first-order solving technique.
They can be integrated directly to obtain the result
Thus integrating w.r.t x
7. First order Linear Differential form
The general format of the equation is in the form
This is a must format and the technique could be applied only when this form is satisfied.
Now we need to find the Integration Factor (I.F.)=
Now multiply each term with the I.F.
Hence we get
Now by exact form, the LHS can be converted as
Now integrate both sides to obtain the required result.
Thus our next form becomes
Thus our equation becomes
Thus solution becomes
x=ye2x + C
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