Please disable adblock to view this page.

← Go home

Differential Equations

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Order of differential equation

Definition: The order of differential equation is the order of highest derivation that appears in the equation

Degree of differential equation

Definition: It is the degree of highest order differential coefficient or derivative, when differential coefficients are free from fractions

General solution of differential equation

A relation between the dependent and independent variables, which is free from derivatives, which satisfies a given differential equation which contains arbitrary constants equal to the order of the differential equation is called general solution or complete integral.

Particular solution of a differential equation

The solution obtained by assigning particular values to the arbitrary constants in G.S. of a differential equation is called a particular equation or particular integral

Ordinary Differential Equation of First Order and First Degree

Form : M dx + N dy = 0

where, M and N are functions of x and y or constants

Method of solving depends on following types:

Variables Separable Form (V.S.)

Reduce to form: $g(y) dy = f(x) dx$ or ${dy}/{g(y)} = {dx}/{f(x)}$



Solution: Integrate both sides e.g.
$∫g(y) dy = ∫f(x) dx + C$

Differential equations reducible to V.S. form by using substitution

1. Linear Solution

Form: ${dy}/{dx} = f(ax + by + c)$
Substitute: ax + by + c = u




2. Quotient Substitution
(i) Form: ${dy}/{dx} = f({y}/{x})$

Substitute: $y/x = u$

Thus, ${dy}/{dx} = u + x {du}/{dx}$

(ii) Form: ${dx}/{dy} = f ({x}/{y})$

Substitute: $x/y = u$
Thus, ${dx}/{dy} = u + y {du}/{dy}$

Homogeneous Differential Equation

Form: M(x, y) dx + N(x, y) dy = 0 or ${dx}/{dy} = {M(x,y)}/{N(x,y)}$

In this, the degree of M and N functions in x and y, should be same

Strategy: Reduce to V.S. form by changing the dependent variable from y to u by substituting y = ux

Thus, ${dy}/{dx} = u + x {du}/{dx}$

Non-Homogeneous Differential Equation reducible to Homogeneous Form

Form: ${dx}/{dy} = {a_1 x + b_1 y +c_1}/{a_2 x + b_2 y + c_2}$

Case(i): $a_1/a_2 = b_1/b_2$

In this case, the expressions $a_1 x + b_1 y$ and $a_2 x + b_2 y$ will always have a common factor of the form lx + my.
Put lx+my = u, then equation reduces to V.S. form in the variables u, x.

Case(ii): $a_1/a_2 ≠ b_1/b_2 $

Substitution: x = X + h and y = Y + k (h and k are constants)

Thus, ${dY}/{dX} = {a_1 X + b_1 Y + (a_1 h + b_1 k + c_1)}/{a_2 X + b_2 Y + (a_2 h + b_2 k + c_2)}$

Consider, $a_1 h + b_1 k +c_1 = 0$ and $a_2 h + b_2 k + c_2 = 0$ to find h and k

We have, ${dY}/{dX} = {a_1 X + b_1 Y }/{a_2 X + b_2 Y}$

It’s a homogenous equation in X and Y. Put Y = VX, ${dY}/{dX} = V + X {dV}/{dX}$.

Finally equation reduces to V.S. form

Exact Differential Equations

Form: M dx + N dy = du , where u(x,y)

Condition of Exactness

${δM}/{δy} = {δN}/{δx}$

Rules if condition is satisfied:

Rule 1: $∫_{y=constant} M dx + ∫ [\text"Terms of N not containing x"] dy = C$

Rule 2: If N has no term which is free from x, $∫_{y=constant} M dx = C$ is general solution

Rule 3: Sometimes we may write the G.S. by using the following rule:

$∫_{x=constant} N dy + ∫ [\text"Terms of M not containing y"] dx = C$

Remark: Sometimes equation of form ${dx}/{dy} = {a_1 x + b_1 y +c_1}/{a_2 x + b_2 y + c_2}$ becomes exact if $b_1 = – a_2$ i.e.

Solution: $∫_{y=constant} (a_1 x + b_1 y + c_1) dx – ∫ (b_2 y + c_2) dy = C$

Equations reducible to exact form by using integrating factor

I.F. is a multiplying factor by which the equation becomes exact



Rule 1: x M + y N ≠ 0 and then given equation is homogeneous, then

I.F. = ${1}/{x M + y N}$


Rule 2: if x M – y N ≠ 0 and given D.E. has,

Form : $y* f_1 (xy) dx + x*f_2(xy) dy = 0$ then
I.F. = ${1}/{x M – y N}$

Rule 3: If ${{δM}/{δy} – {δN}/{δx}}/{N}$ = f(x) [say] then

I.F. = $e^{∫f(x) dx}$

Rule 4: If ${{δN}/{δx} – {δM}/{δy}}/{M}$ = f(y) [say] then

I.F. = $e^{∫f(y) dy}$

Rule 5: If equation M dx + N dy = 0 can be written as:

$x^a y^b (my dx + nx dy) + x^r y^s (py dx + qx dy) = 0$

where, a, b, m, n, r, s, p, q are constants then
I.F. = $x^h y^k$

Note: h, k can be determined from the following two equations:

nh – mk = (m-n) + (mb-na)

qh – pk = (p-q) + (ps – qr)

Integrating Factors found by inspection

x dy + y dx = d(xy)

${xdy + y dx}/{xy} = d (log(xy))$

${xdy – y dx}/{x^2} = d(y/x)$

${x dy – y dx}/{xy} = d [log(y/x)]$

${x dy – y dx}/{x^2 + y^2} = d[tan^{-1}(y/x)]$

${x dy – y dx}/{x^2 – y^2} = d (1/2 log {x+y}/{x-y})$

${y dx – x dy}/{y^2} = d (x/y)$

${y dx – x dy}/{x^2 + y^2} = d(tan^{-1}(x/y))$

${y dx – x dy}/{xy} = d (log(x/y))$

${x dx + y dy}/{x^2 + y^2} = 1/2 d (log(x^2 + y^2))$

${x dx + y dy}/{√{x^2 + y^2}} = d(√{x^2 + y^2})$

$x dx + y dy = 1/2 d (x^2 + y^2)$

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

${dx + dy}/{x + y} = d log (x+y)$

$(x+y)^n (dx + dy) = d [{(x+y)^{n+1}}/{n+1}] if n≠ -1$

${x dy + y dx}/{x^2 y^2} = d({-1}/{xy})$

${y 2x dx – x^2 dy}/{y^2} = d ({x^2}/{y})$

${x 2y dy – y^2 dx}/{x^2} = d ({y^2}/{x})$

${2x^2 y dy – 2 y^2 x dx}/{x^4} = d (y^2 / x^2)$

${2xy^2 dx – 2 yx^2 dy}/{y^4} = d(x^2/y^2)$

${y e^x dx – e^x dy}/{y^2} = d({e^x}/{y})$


Linear Differential Equations of the First Order

Form: ${dy}/{dx} + Py = Q$ where, P & Q are functions of ‘x’ or constants

Method of Solution
I.F. = $e^{∫P dx}$


G.S. = $y e ^{∫P dx} = ∫ Q . e^{∫P dx} dx + C$

Similarly, ${dx}/{dy} + P x = Q$ , where P and Q are functions of ‘y’ or constants

Method of Solution
I.F. = $e^{∫P dy}$


G.S. = $x e ^{∫P dy} = ∫ Q . e^{∫P dy} dy + C$