JEE Maths›Differential Equations›Question Types

Differential Equations - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Find Order and Degree

Pattern

Determine the order and degree of a given differential equation

How to recognize

A differential equation is given and you are asked for its order, degree, or both

→ Identify the highest order derivative for order. Express the DE as a polynomial in the highest order derivative for degree. If it cannot be expressed as a polynomial, degree is not defined.

Solve Variable Separable DE

Pattern

Separate variables and integrate both sides

How to recognize

dy/dx can be written as a product f(x)*g(y) or a ratio h(x)/k(y)

→ Rearrange to get all y terms with dy on one side and all x terms with dx on the other. Integrate both sides. Do not forget g(y) = 0 cases when dividing.

Solve Homogeneous DE

Pattern

Use y = vx substitution for homogeneous equations

How to recognize

dy/dx = F(y/x), or each term has the same total degree in x and y, or can be written as a function of y/x

→ Put y = vx, so dy/dx = v + x(dv/dx). Substitute, separate variables in v and x, integrate, then replace v = y/x.

Solve Linear First-Order DE

Pattern

Find integrating factor and apply the linear DE formula

How to recognize

DE can be written as dy/dx + P(x)y = Q(x) where P and Q are functions of x only (or the dx/dy form)

→ Identify P(x) and Q(x). Compute IF = e^(integral of P dx). Multiply both sides by IF. The left side becomes d/dx(y * IF). Integrate the right side.

Form DE from Family of Curves

Pattern

Eliminate arbitrary constants by differentiation

How to recognize

A family of curves with one or more arbitrary constants is given; find its differential equation

→ Count the number of arbitrary constants (n). Differentiate the equation n times. Use the original equation plus n derived equations to eliminate all n constants.

Find Orthogonal Trajectory

Pattern

Replace dy/dx with -dx/dy in the family's DE and solve

How to recognize

Find the curves that intersect a given family of curves at right angles

→ First find the DE of the given family (eliminate constants). Replace dy/dx with -dx/dy. Solve the resulting DE to get the orthogonal trajectories.

Application Problems (Growth/Decay/Cooling)

Pattern

Set up and solve a DE from a word problem involving rates

How to recognize

Problems involving population growth, radioactive decay, cooling, mixing, or any quantity changing at a rate proportional to itself

→ Translate the word problem into dN/dt = kN or dT/dt = -k(T - T_s). Solve the DE. Use initial/boundary conditions to find constants.

Find Particular Solution with Initial Condition

Pattern

Solve the DE and use the given initial condition to find the constant

How to recognize

A DE is given along with a condition like y(0) = 1 or the curve passes through a specific point

→ First find the general solution (with constant C). Substitute the initial condition to determine C. Write the particular solution.

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Application of Derivatives

Conic Sections

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