Revision: Differential Equations Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

An equation involving independent variable(s), dependent variable(s), derivatives of the dependent variable (s) with respect to the independent variable(s), and a constant is called a differential equation.

The order of the highest differential coefficient (or the highest order derivative appearing in a differential equation) is the order of the differential equation.

The highest exponent of the highest derivative is called the degree of a differential equation, provided exponents of each derivative and an unknown variable appearing in the differential equation are non-negative integers.

A differential equation of the form \[\frac{dy}{dx}=\frac{f_{1}(x,y)}{f_{2}(x,y)},\] where f₁(x, y) ) and f₂(x, y)  are homogeneous functions of x and y of the same degree, is called a homogeneous differential equation.

A linear differential equation of first order and first degree is
\[\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{P}y=\mathrm{Q}\], where P and Q are the functions of x or constants. Its general solution is  \[y.\left(\mathrm{I.F.}\right)=\int\mathrm{Q.}\left(\mathrm{I.F.}\right)\mathrm{d}x+\mathrm{c}\] and the function \[\mathrm{e}^{\int\mathrm{Pdx}}\] is called the integrating factor (I.F.) of the given equation.

A solution of a differential equation in which the number of arbitrary constants equals the order of the differential equation is called the general solution of the differential equation.

A solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution.

Any relation between independent and dependent variables which does not involve derivatives, such that this relation and the derivatives obtained from it satisfy the given differential equation, is called a solution of the differential equation.

A function f(x,y) is called a homogeneous function of degree n if the degree of each term is n.

A differential equation is non-linear if any one of the following holds:

1. The degree is more than one

2. Any differential coefficient has an exponent of more than one

3. Exponent of the dependent variable is more than one

4. Products containing the dependent variable and its differential coefficients are present

A solution or an integral of a differential equation is a function of the form y = f(x) which satisfies the given differential equation. 

A first-order differential equation, along with an initial condition, is called an initial value problem.

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

General Form: \[\frac{dy}{dx}+Py=Q\]

where P and Q are functions of x.

A differential equation of the form \[\frac{dy}{dx}=\frac{f(x,y)}{\phi(x,y)}\] where f(x,y) and ϕ(x,y) are homogeneous functions of the same degree, is called a homogeneous differential equation.

A differential equation is an equation that involves independent and dependent variables and their derivatives.

The order of a differential equation is the order of the highest derivative occurring in it.

The degree of a differential equation is the degree of the derivative of the highest order occurring in it after the equation is freed from radical signs and fractions in the derivative. 

General Solution

• A solution containing arbitrary constants equal to the order of the differential equation is called the general solution.

Particular Solution

• Solutions obtained by giving particular values to the arbitrary constants in the general solution are called particular solutions.

A differential equation in which the variables can be separated is of the form

\[f(x)dx+\phi(y)dy=0\]

The factor e^∫P dx on multiplying by which the left-hand side of the differential equation\[\frac{dy}{dx}+Py=Q\] becomes the differential coefficient of a function of x and y, is called the integrating factor of the differential equation.

General Solution: \[y\cdot\mathrm{I.F.}=\int Q\cdot\mathrm{I.F.}dx+c\]

Since ‘a’ lies between 0 and 2a,
we have

`int_0^(2a)f(x)dx=int_0^af(x)dx+int_a^(2a)f(x)dx,  .......(byint_a^bf(x)dx=int_a^cf(x)dx+int_c^bf(x)dx)`

`=I_1+I_2`     ........................(say)

`I_2 = int_a^(2a)f(x)dx`

Put x = 2a − t

Therefore, dx = −dt

When x = a, 2a − t = a

t = a

When x = 2a, 2a − t = 2a

t = 0

`I_2 = int_0^(2a) f(x) dx = int_a^0 f(2a - t) (-dt)`

`= -int_a^0 f(2a - t)dt = int_0^a f(2a - t)dt      ...................... (By int_a^b f(x)dx = -int_b^a f(x)dx)`

`=int_0^a f(2a - x)dx    ..............(By int_a^b f(X)dx = int_a^b f(t)dt)`

`int_0^(2a) f(x)dx = int_0^a f(x)dx + int_0^a f(2a - x)dx`

`= int_0^a [f(x) + f(2a - x)]dx`

To show that:

`int_0^pi sin x  dx = 2 int_0^(pi/2) sin x  dx`

We use the proven property by setting f(x) = sin x and 2a = π, which means a = `pi/2`.

The property tells us that:

`int_0^pi sin x  dx = int_0^(pi/2) sin  x  dx + int_0^(pi/2) sin (pi - x)  dx`

Knowing the trigonometric identity sin (π - x) = sin x, the equation simplifies to:

`int_0^pi sin x  dx = 2 int_0^(pi/2) sin x  dx`

This directly applies the property to the integral of sin x over [0, π] to show it equals twice the integral of sin x over `[0, pi/2]`, demonstrating the utility of this property in simplifying integrals with symmetric functions over specific intervals​.

1. Basic Idea:

• Form a differential equation from a given equation by eliminating arbitrary constants

2. Steps:

• Identify arbitrary constants in the given equation

• Differentiate the equation with respect to x as many times as the number of constants

• Eliminate constants from the obtained equations

3. Important Rule:

• Number of differentiations = number of arbitrary constants

4. Final Result:

• After eliminating constants → required differential equation is obtained

6. Important Note:

• A differential equation represents a family of curves

(i) Express the homogeneous differential equation in the form
dy/dx = f(x, y) / g(x, y)

(ii) Put y = vx and
dy/dx = v + x dv/dx

Substitute in the equation and cancel out x from the R.H.S.
The equation reduces to the form
v + x dv/dx = F(v)

(iii) Take v on R.H.S. and separate the variables v and x

(iv) Integrate both sides to obtain the solution in terms of v and x

(v) To obtain the required solution in terms of x and y, substitute v = y/x

(i) Write the equation in the form dy/dx + Py = Q

(ii) Identify P and Q

(iii) Find I.F. = \[\mathrm{e}^{\int\mathrm{Pdx}}\]

(iv) Multiply the whole equation by I.F.

(v) Integrate and get a solution.

1. Population Growth

• Rate of change of population ∝ population

• \[\frac{\mathrm{dP}}{\mathrm{dt}}=\mathrm{kP}\]

Growth increases with time

2. Radioactive Decay

• Rate of decay ∝ of the amount present

• \[\frac{\mathrm{d}x}{\mathrm{d}t}=-\mathrm{k}x\]

Negative sign → quantity decreases

3. Newton’s Law of Cooling

• Rate of cooling ∝ temperature difference

• \[\frac{\mathrm{d}\theta}{\mathrm{d}t}=-k\left(\theta-\theta_{0}\right)\]

θ = body temp, θ₀ = surrounding temp

1. Put y = vx

2. Separate the variables v and x

3. Integrate both sides

4. Replace v by \[\frac{y}{x}\]​

1. Radioactive Decay:  \[x=x_0e^{-kt}\]

2. Half-Life Formula: \[k=\frac{\ln2}{T}\]

3. Newton’s Law of Cooling: \[\theta=\theta_0+(\theta_1-\theta_0)e^{-kt}\]

4. Population Growth: \[P=ae^{kt}\]

• Write the equation in the form
  \[\frac{dy}{dx}+Py=Q\]

• Find the integrating factor
  \[\mathrm{I.F.}=e^{\int Pdx}\]

• Multiply the entire equation by I.F.

• Integrate both sides w.r.t x

• Obtain
  \[y(\mathrm{I.F.})=\int Q(\mathrm{I.F.})dx+c\]