Revision: Diffrential Equations JEE Main Diffrential 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.

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 containing arbitrary constants is called the general solution of a 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.

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.

• 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.

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

• 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.