Maths > Differential Equations > 1.0 Introduction
Differential Equations
1.0 Introduction
2.0 Methods to find the solution of first order and first degree differential equation
2.1 Variable separable method
2.2 Equations reducible to variable separable form
2.3 Homogeneous differential equation
2.4 Equations reducible to homogeneous form
2.5 Exact differential equation
2.6 Equations reducible to exact form
2.7 Linear differential equation
2.8 Equations reducible to linear form
3.0 Differential equation of first order and higher degrees
4.0 Orthogonal trajectory
1.2 Formation of a differential equation
2.2 Equations reducible to variable separable form
2.3 Homogeneous differential equation
2.4 Equations reducible to homogeneous form
2.5 Exact differential equation
2.6 Equations reducible to exact form
2.7 Linear differential equation
2.8 Equations reducible to linear form
A differential equation corresponding to a family of curve must have order exactly equal to the number of arbitrary constants in the equation of
curve and no arbitrary constant in it. In order to remove the constants and form a differential equation, following steps are to be followed:
1. Identify the number of arbitrary constants in given equation of curve.
2. Differentiate the given equation with respect to the independent variable as many times as the number of arbitrary constants in it.
3. Eliminate the arbitrary constants and eliminant is the required differential equation.
Question 2. Form the differential equation, if ${y^2} = 4a(x + b)$, where $a$, $b$ are arbitrary constants.
Solution: Since, there are two arbitrary constants in the given equation of curve, we need to differentiate two times with respect to $x$ to remove them.
Differentiate with respect to $x$, we get $$\begin{equation} \begin{aligned} 2y\frac{{dy}}{{dx}} = 4a \\ y\frac{{dy}}{{dx}} = 2a \\\end{aligned} \end{equation} $$
Again differentiate with respect to $x$, we get $$y\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^2} = 0$$ which is the required differential equation.
Question 3. Form a differential equation of family of circles touching $x$-axis at the origin.
Solution:
Equation of family of circles touching $x$-axis at origin can be written as $${x^2} + {y^2} + \lambda y = 0\quad ...(1)$$ where $\lambda $ is a parameter. Differentiate w.r.t. $x$, we get $$2x + 2y\frac{{dy}}{{dx}} + \lambda \frac{{dy}}{{dx}} = 0$$ Put the value of $\lambda $ from $(1)$, we get $$ 2x + 2y\frac{{dy}}{{dx}} + \left( { - \frac{{{x^2}}}{y} - y} \right)\frac{{dy}}{{dx}} = 0 $$ $$ \Rightarrow 2x + 2y\frac{{dy}}{{dx}} - \frac{{{x^2}}}{y}\frac{{dy}}{{dx}} - y\frac{{dy}}{{dx}} = 0 $$ $$ \Rightarrow 2x + \frac{{dy}}{{dx}}\left( {y - \frac{{{x^2}}}{y}} \right) = 0 $$ $$ \frac{{dy}}{{dx}} = \frac{{2xy}}{{{x^2} - {y^2}}} $$
which is the required differential equation.
