DIFFERENTIAL EQUATION:

• An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a DIFFERENTIAL EQUATION.

• Finding the unknown function which satisfies given differential equation is called SOLVING OR INTEGRATING the differential equation.

• The solutionof the differential equation is also called its PRIMITIVE, because the differential equation can be regarded as a relation derived from it.

ORDER OF DIFFERENTIAL EQUATION:

$\quad$ The order of a differential equation is the order of the highest differential coefficient occurring in it.

DEGREE OF DIFFERENTIAL EQUATION :

$\quad$ The degree of a differential equation which can be written as a polynomial in the derivatives is the degree of the derivative of the highest order occurring in it,

$\quad$ after it has been expressed in a form free from radicals & fractions so far as derivatives are concerned, thus the differential equation :

$\quad$ The differential equation

$$f(x, y)\left[\frac{d^{m} y}{d x^{m}}\right]^{p}+\phi(x, y)\left[\frac{d^{m-1} y}{d x^{m-1}}\right]^{q}+\ldots+ C=0$$

$\quad$ is of order $m$ and degree $p$.

$\quad$ Note that in the differential equation $e^{y^{\prime \prime}}-x y^{\prime \prime}+y=0$ order is three but degree doesn't exist.

FORMATION OF A DIFFERENTIAL EQUATION :

• If an equation in independent and dependent variables having some arbitrary constant is given, then a differential equation is obtained as follows :

  • Differentiate the given equation w.r.t the independent variable (say $\mathrm{x}$ ) as many times as the number of arbitrary constants in it.

  • Eliminate the arbitrary constants.The eliminant is the required differential equation.

  Note : A differential equation represents a family of curves all satisfying some common properties. This can be considered as the geometrical interpretation of the differential equation.

General And Particular Solutions:

• The solution of a differential equation which contains a number of independent arbitrary constants equal to the order of the differential equation is called the GENERAL SOLUTION (OR COMPLETE INTEGRAL OR COMPLETE PRIMITIVE).

• A solution obtainable from the general solution by giving particular values to the constants is called a PARTICULAR SOLUTION.

ELEMENTARY TYPES OF FIRST ORDER \& FIRST DEGREE DIFFERENTIAL EQUATIONS :

PYQ-2023-DE-Q5, PYQ-2023-Area_Under_Curves-Q12

• Variables separable:

  $\quad$ TYPE-1

  $\quad$ If the differential equation can be expressed as $$f(x) dx+g(y) d y=0$$ $\quad$ then this is said to be variable separable type.

  $\quad$ A general solution of this is given by $$\int f(x) , dx + \int g(y) , dy = c$$ $\quad$ is the arbitrary constant.

  $\quad$ TYPE-2

  $\quad$ The differential equation

  $$ \frac{dy}{dx} = f(ax + by + c), \quad b \neq 0 $$

  $\quad$ can be solved by substituting

  $$ t = ax + by + c. $$

  $\quad$ Then the equation reduces to a separable type in the variables $t$ and $x$, which can be solved.

• Homogeneous equations: PYQ-2023-DE-Q2 PYQ-2023-DE-Q8 PYQ-2023-DE-Q9

  $\quad$ A differential equation of the form

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

  $\quad$ where $f(x, y) \ \text{and} \ \phi(x, y)$ are homogeneous functions of $x \ \& \ y$ and of the same degree, is called HOMOGENEOUS.

  $\quad$ This equation may also be reduced to the form

  $$ \frac{dy}{dx} = g\left(\frac{x}{y}\right) $$

  $\quad$ and is solved by putting $y = vx$ so that the dependent variable $y$ is changed to another variable $v$, where $v$ is some unknown function. The differential equation is transformed to an equation with variables separable.

• Equations reducible to the homogeneous form :

  $\quad$ If $$\frac{dy}{dx}=\frac{a_1 x+b_1 y+c_1}{a_2 x+b_2 y+c_2}$$

  $\quad$ where $a_1 b_2-a_2 b_1 \neq 0$, i.e. $\frac{a_1}{b_1} \neq \frac{a_2}{b_2}$ then

  $\quad$ the substitution $\mathrm{x}=\mathrm{u}+\mathrm{h}, \mathrm{y}=\mathrm{v}+\mathrm{k}$ transform this equation to a homogeneous type in the new variables $\mathrm{u}$ and $\mathrm{v}$ where $\mathrm{h}$ and $\mathrm{k}$ are arbitrary constants to be chosen so as to make the given equation homogeneous.

  • $\quad$ If $\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1=0$, then a substitution $\mathrm{u}=\mathrm{a}_1 \mathrm{x}+\mathrm{b}_1 \mathrm{y}$ transforms the differential equation to an equation with variables separable.

  • $\quad$ If $b_1+a_2=0$, then a simple cross multiplication and substituting $d(xy)$ for $x dy+y dx \&$ integrating term by term yields the result easily.

  • $\quad$ In an equation of the form : $y f(xy) dx+x g(xy) dy=0$ the variables can be separated by the substitution $xy=v$.

LINEAR DIFFERENTIAL EQUATIONS:

PYQ-2023-DE-Q1, PYQ-2023-AOD-Q12

$\quad$ A differential equation is said to be linear if the dependent variable $\&$ its differential coefficients occur in the first degree only and are not multiplied together.

$\quad$ The nth order linear differential equation is of the form;

$$a_0(x) \frac{d^n y}{dx^n}+a_1(x) \frac{d^{n-1} y}{dx^{n-1}}+\ldots \ldots \ldots+a_n(x)$$

$\quad y=\phi(x)$, where $a_0(x), a_1(x) \ldots a_n(x)$ are called the coefficients of the differential equation.

• Linear Differential Equations Of First Order : PYQ-2023-DE-Q4 PYQ-2023-DE-Q6 PYQ-2023-DE-Q7 PYQ-2023-DE-Q10

  The most general form of a linear differential equations of first order is $\frac{dy}{dx}+P y=Q$, where $P \& Q$ are functions of $x$.

  To solve such an equation multiply both sides by $e^{\int P dx}$. Then the solution of this equation will be $y e^{\int \mathrm{Pdx}}=\int \mathrm{Q} e^{\int \mathrm{Pdx}} \mathrm{dx}+\mathrm{c}$

• Equations Reducible To Linear Form : PYQ-2023-DE-Q3 PYQ-2023-DE-Q11

  The equation $\frac{dy}{dx}+P y=Q \cdot y^n$ where $P \& Q$ are function, of $x$, is reducible to the linear form by dividing it by $y^n \&$ then substituting $y^{-n+1}=Z$.

  The equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{Py}=\mathrm{Q} \mathrm{y}^{\mathrm{n}}$ is called BERNOULI’S EQUATION.

Trajectories :

$\quad$ A curve which cuts every member of a given family of curves according to a given law is called a Trajectory of the given family.

• Orthogonal trajectories :

  $\quad$ A curve making at each of its points a right angle with the curve of the family passing through that point is called an orthogonal trajectory of that family.

  $\quad$ We set up the differential equation of the given family of curves. Let it be of the form $\mathrm{F}\left(\mathrm{x}, \mathrm{y}, \mathrm{y}^{\prime}\right)=0$

  $\quad$ The differential equation of the orthogonal trajectories is of the form

  $$ F\left(x, y, \frac{-1}{y^{\prime}}\right)=0 $$

  $\quad$ The general integral of this equation $\phi_1(x, y, C)=0$ gives the family of orthogonal trajectories.

Important Notes:

$$x dy+y dx=d(xy)$$

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

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

$$\frac{x dy+y dx}{xy}=d(xy )$$

$$\frac{dx+dy}{x+y}=d(\ln (x+y))$$

$$\frac{x dy-y dx}{xy}=d(\ln \frac{y}{x})$$