Revision: 12th Std >> Differential Equations MAH-MHT CET (PCM/PCB) Differential Equations

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

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

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.

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.

The equation \[\frac{dy}{dx}=f(x,y)\] is to be in variable separable form if it can be expressed as \[h(x)dx=g(y)dy\].

The solution to this equation is obtained by integrating h(x) and g(y) with respect to x and y, respectively.

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.

Equations of the form\[\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{P}y=\mathrm{Q}y^{\mathrm{n}}\]where (P) and (Q) are functions of (x) is called Bernoulli’s equation.

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