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Kerala Plus Two Maths Notes Chapter 9 Differential Equations
Introduction
An equation involving derivatives of a dependent variable with respect to one or more independent variables is called a Differential Equation. In this chapter we study the method formation of a Differential Equation and solving of a Differential Equation.
I. Degree and Order of a DE
Order of a DE is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given DE.
Degree of a DE is defined as the exponent of highest differential coefficient appearing in the equation provided the equation is made into polynomial form in all differential coefficient.
II. Formation of a DE
To form a DE from a given function we differentiate the function successively as many times as the number of arbitrary constants in the equation and eliminate the arbitrary constant.
III. Solution of a DE
1. Variable Separable Type:
A DE of the form mdx = ndy Where m is a function in x alone or a constant and n is a function y alone or a constant.
Solution is ∫mdx = ∫ndy + c.
2. Homogeneous DE:
A DE of the form \(\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)}\), where f(x, y) and g(x, y) are homogeneous equations in x and y. Solution is put y = vx ⇒ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) after simplification DE will be converted into variable separable type.
3. Linear DE:
A DE of the form \(\frac{d y}{d x}\) + Py = Q, where P and Q dx are function in x alone or a constant.
Solution is IF = e∫Pdx
⇒ y(IF) = ∫Q(IF)dx + c.
A DE of the form \(\frac{d x}{d y}\) + px = Q, where P and Q are function in y alone or a constant.
Solution is IF = e∫Pdy
⇒ x(IF) = ∫Q(IF)dy + c.