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

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

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.

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