Answer

Verified

415.2k+ views

**Hint:**A differential equation(D.E.), as the name suggests, is an equation consisting of one or more terms and derivative(s) of the dependent variable with respect to the independent variable.

When an equation for the curve is given, first of all we have to see whether the terms of dependent and the independent variables can be separated successfully after differentiating the given equation or not? The answer to this question will make our decision to use one of the following methods:

a) Variable separable method

b) Integrating factor method

**Complete step-by-step answer:**

a) The given curve is $y = c\left( {x - 2c} \right)$

It can be re-written as $y = cx - 2{c^2}$ ……………….. Eqn I

Here, the independent variable is $x$ and the dependent variable is $y$.

Differentiating w.r.t. $x$, we get

$\dfrac{{dy}}{{dx}} = \dfrac{{d\left( {cx} \right)}}{{dx}} - \dfrac{{d\left( {2{c^2}} \right)}}{{dx}}$

$ \Rightarrow \dfrac{{dy}}{{dx}} = c\dfrac{{d\left( x \right)}}{{dx}} - 0$ $\left[ {\because \dfrac{{d\left( {const} \right)}}{{dx}} = 0} \right]$

$ \Rightarrow \dfrac{{dy}}{{dx}} = c$ ……………….. Eqn II

Using the value of the parameter $c$ from Eqn II and putting in Eqn I, we have

$y = x\dfrac{{dy}}{{dx}} - 2{\left( {\dfrac{{dy}}{{dx}}} \right)^2}$ …………………… Eqn A

This is required D.E. for the curve $y = c\left( {x - 2c} \right)$, where $c$ is a parameter.

b) The given curve is $y = cx + c - {c^2}$ …………………….. Eqn I

Here, the independent variable is $x$ and the dependent variable is $y$.

Differentiating w.r.t. $x$, we get

\[\dfrac{{dy}}{{dx}} = \dfrac{{d\left( {cx} \right)}}{{dx}} + \dfrac{{d\left( c \right)}}{{dx}} - \dfrac{{d\left( {{c^2}} \right)}}{{dx}}\]

$ \Rightarrow \dfrac{{dy}}{{dx}} = c\dfrac{{d\left( x \right)}}{{dx}} + 0 - 0$ $\left[ {\because \dfrac{{d\left( {const} \right)}}{{dx}} = 0} \right]$

$ \Rightarrow \dfrac{{dy}}{{dx}} = c$ …………….. EqnII

Using the value of the parameter $c$ from Eqn II and putting in Eqn I, we have

$y = x\dfrac{{dy}}{{dx}} + \dfrac{{dy}}{{dx}} - {\left( {\dfrac{{dy}}{{dx}}} \right)^2}$

$y = \left( {x + 1} \right)\dfrac{{dy}}{{dx}} + {\left( {\dfrac{{dy}}{{dx}}} \right)^2}$ …………………….. Eqn B

This is required D.E. for the curve $y = cx + c - {c^2}$, where $c$ is a parameter.

**Note:**Both Eqn A and Eqn B are First order D.E. because the highest order of derivatives in both the cases is $1$. [as we can see $\left( {\dfrac{{dy}}{{dx}}} \right)$ only and no $\dfrac{{{d^2}y}}{{d{x^2}}}$……..].DEs are important for the fact that we can compute the function on the entire domain. Higher studies in the fields of Chemistry, Biology, Physics, Economics and so on use mathematical models to predict or analyse things. There comes the main use of the DE.

Recently Updated Pages

Which are the Top 10 Largest Countries of the World?

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Name 10 Living and Non living things class 9 biology CBSE

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE