Revision: Mathematics >> Differential Equations CUET (UG) Differential Equations

If an equation contains derivatives of one dependent variable with respect to one or more independent variables, then it is called a differential equation.

Example

This is a differential equation because it contains the derivative \[\frac{dy}{dx}\].

A differential equation that contains ordinary derivatives of one or more dependent variables with respect to a single independent variable is called an ordinary differential equation.

Example:

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.

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

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

For a differential equation, a solution is a function that makes the left-hand side equal to the right-hand side when the function and its required derivatives are substituted. If y = ϕ(x) satisfies the differential equation, then the curve represented by y = ϕ(x) is called the solution curve or integral curve.

A solution containing arbitrary constants is called the general solution of a differential equation.

The equation \[\frac{dy}{dx} = F(x, y)\] is said to be in variable separable form if it can be expressed as \[g(x) dx = h(y) dy\] or equivalently as \[\frac{dy}{dx} = g(x)h(y)\] so that the variables can be separated and integrated.

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.

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​.

• A differential equation contains derivatives.

• An ordinary differential equation contains derivatives with respect to only one independent variable.

• Differential equations describe rates of change in mathematics and science.

• Order = highest derivative order.

• Degree = power of highest derivative.

• Degree exists only for polynomial equations in derivatives.

• Always check polynomial condition before stating the degree.

• A differential equation contains derivatives of an unknown function.

• Its solution is generally a function, not a single number.

• The graph of the solution function is called the solution curve or integral curve.

• A general solution contains arbitrary constants.

• A particular solution is obtained by assigning fixed values to those constants.

• To verify a solution, substitute the function and its derivatives into the equation and check whether LHS = RHS.

• Variable separable equations can be rewritten as x-part = y-part.

• Separate variables first, then integrate.

• Use one constant of integration.

• Apply the initial condition only after getting the general solution.

• Final answers may be explicit or implicit.

• Check homogeneity first.

• Differentiate substitution carefully.

• Convert to separable form.

• Back-substitute to original variables.

1. Write the equation in the form dy/dx + Py = Q
2. Identify P and Q
3. Find I.F. = \[\mathrm{e}^{\int\mathrm{Pdx}}\]
4. Multiply the whole equation by I.F.
5. Integrate and get a solution.