Important Formula and Notes in Differential Equations

Dear students, before you start revision for the chapter Differential Equations, first make sure you revise all the formulas and key points. Only if you are thorough with the main concepts and formulas, you can confidently work out all the sums. That said, here are the most important formulas in Differential Equations, Class 12 Business Maths.

Important Formula and Notes in Differential Equations

Family of Curves

(i) y² = 4ax represents the equation of a family of parabolas having the origin as vertex where ‘a’ is the parameter.
(ii) x² + y² = a² represents the equation of family of circles having the origin as centre, where ‘a’ is the parameter.
(iii) y = mx + c represents the equation of a family of straight lines in a plane, where m and c are parameters.

Order of the highest order derivative present in the differential equation is the order of the differential equation.

Degree is the highest power of the highest order derivative in the differential equation, after the equation has been cleared from fractions and the radicals as for as the derivatives are concerned.

A differential equation of the form dy/dx + Py = Q where P and Q are constants or functions of x only is called a first order linear differential equation.

A function which satisfies the given differential equation is called its solution.

The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution.

A solution free from arbitrary constants is called particular solution.

To form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants.

A general second order linear differential equation with constant coefficients is of the form:

$a \dfrac{d^{2}y}{dx^{2}}+b \dfrac{dy}{dx}+cy= f(x)\\$

A differential equation of the form

$\frac{dy }{dx}=\frac{f(x,y)}{g(x,y)} \$

is called homogeneous differential equation if f (x, y) and g(x, y) are homogeneous functions of the same degree in x and y.

$If \;the \;problem \;gives\; \frac{dy}{dx}\; put\;y=vx\;\Rightarrow v=\frac{y}{x} \\$
$If \;the \;problem \;gives\; \frac{dx}{dy}\; put\;x=vy\;\Rightarrow v=\frac{x}{y} \\$

Second Order first degree differential equations with constant coefficients

Real \;and \;different\;(m_{1}\neq m_{2} ) \

$Particular \;Integral\;P.I=\frac{1}{\phi(D)}f(x)\\$
$The \; general \;solution \;is\; y=C.F+P.I\; \\$

You can find links to the other exercises below.

Text Book Solutions for Differential Equations Exercise 4.1

Text Book Solutions for Differential Equations Exercise 4.2

Text Book Solutions for Differential Equations Exercise 4.3

Text Book Solutions for Differential Equations Exercise 4.4

Text Book Solutions for Differential Equations Exercise 4.5

Text Book Solutions for Differential Equations Exercise 4.6

Case	Nature of the roots	Complementary Function
1	$Real \;and \;different\;(m_{1}\neq m_{2} ) \$	$Ae^{m_{1}x}+Be^{m_{2}x}\$
2	$Real \;and \;equal\;(m_{1}\=m_{2} )=m \$	$\left( Ax+B \right)e^{mx} \$
3	$Complex\; roots\;(\alpha \pm \beta) \$	$e^{ax} \left( Acos \beta x+Bsin \beta x \right) \$

Important Formula and Notes in Differential Equations

Important Formula and Notes in Differential Equations

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