Differential Equations formulas for JEE Main & Advanced

19 formula entries for Differential Equations, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Variable Separable Method

\int f (x) d x = \int g (y) d y + C

When to use. When DE can be written as $\frac{d y}{d x} = h (x) \cdot k (y)$ .

Trap. Don't forget the constant of integration; it's crucial for general solution.

Homogeneous Differential Equation

y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}

When to use. When DE is of the form $\frac{d y}{d x} = \frac{P (x, y)}{Q (x, y)}$ where P and Q are homogeneous of same degree.

Trap. First check homogeneity: each term in numerator and denominator of $f$ must have same degree.

Linear Differential Equation (First Order)

IF =

e^{\int P d x}

, solution:

y \cdot I F = \int Q \cdot I F d x + C

When to use. When DE is linear in y and its derivative, i.e., no $y^{2}$ or $y y^{'}$ terms.

Trap. Ensure DE is in standard form; coefficient of $\frac{d y}{d x}$ must be 1.

Bernoulli's Equation

v = y^{1 - n} \Rightarrow \frac{d v}{d x} + (1 - n) P v = (1 - n) Q

When to use. When DE has a $y^{n}$ term on RHS, with n ≠ 0,1.

Trap. For n=0 or n=1, it's already linear or separable; don't use Bernoulli substitution.

Exact Differential Equation

Condition:

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

; Solution:

\int M d x + \int (N - \frac{\partial}{\partial y} \int M d x) d y = C

When to use. When DE is in differential form $M d x + N d y = 0$ and condition holds.

Trap. Check exactness condition before solving; if not exact, look for integrating factor.

Integrating Factor for Non-Exact DE

IF =

e^{\int f (x) d x}

e^{\int g (y) d y}

When to use. When DE is not exact but one of the conditions for IF holds.

Trap. Compute partial derivatives correctly; common sign errors.

Orthogonal Trajectories

For family

F (x, y, c) = 0

, find DE, then replace

y^{'}

with

- 1 / y^{'}

When to use. Finding curves perpendicular to a given family (e.g., circles, parabolas).

Trap. First find DE of given family (eliminate constant), then replace slope.

Solving DEs of Form $\frac{d^{2} y}{d x^{2}} = f (x)$

y = \int (\int f (x) d x) d x + C_{1} x + C_{2}

When to use. When the highest derivative is a function of x only.

Trap. Don't forget the two constants of integration.

Reducible to Variable Separable: $\frac{d y}{d x} = f (a x + b y + c)$

v = a x + b y + c \Rightarrow \frac{d v}{d x} = a + b \frac{d y}{d x}

When to use. When DE is of the form $\frac{d y}{d x} = f (a x + b y + c)$ .

Trap. After substitution, solve for $\frac{d y}{d x}$ in terms of $v$ correctly.

General Solution of Linear DE with Constant Coefficients

Roots

m_{1}, m_{2}

: real distinct

\Rightarrow y = C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x}

; real equal

\Rightarrow y = (C_{1} + C_{2} x) e^{m_{1} x}

; complex

α \pm i β \Rightarrow y = e^{α x} (C_{1} \cos β x + C_{2} \sin β x)

When to use. Solving homogeneous linear DEs with constant coefficients (order 2 or higher).

Trap. For repeated roots, multiply by x for second term.

Particular Solution by Method of Undetermined Coefficients

If RHS =

e^{k x}

, guess

A e^{k x}

; if

\sin k x

\cos k x

, guess

A \cos k x + B \sin k x

; if polynomial, guess polynomial of same degree.

When to use. When RHS is a combination of polynomials, exponentials, sines, cosines.

Trap. If guess matches complementary solution, multiply by x (or $x^{2}$ for repeated).

Particular Solution by Variation of Parameters

y_{p} = - y_{1} \int \frac{y_{2} R}{W} d x + y_{2} \int \frac{y_{1} R}{W} d x

When to use. When RHS is not of the form for undetermined coefficients, or for variable coefficient DEs.

Trap. Ensure DE is in standard form (coefficient of $y^{''}$ is 1).

Cauchy-Euler Equation

Roots

m_{1}, m_{2}

: real distinct

\Rightarrow y = C_{1} x^{m_{1}} + C_{2} x^{m_{2}}

; real equal

\Rightarrow y = (C_{1} + C_{2} \ln x) x^{m_{1}}

; complex

α \pm i β \Rightarrow y = x^{α} (C_{1} \cos (β \ln x) + C_{2} \sin (β \ln x))

When to use. When DE has terms like $x^{2} y^{''}$ , $x y^{'}$ , $y$ with constant coefficients.

Trap. For non-homogeneous, use variation of parameters after transforming to constant coefficients via $x = e^{t}$ .

Solving DEs Using Substitution $y = v x$ (Homogeneous) or $x = v y$

y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}

When to use. When DE is homogeneous of degree 0.

Trap. Check homogeneity before substitution.

Reduction of Order

y_{2} = y_{1} \int \frac{e^{- \int P d x}}{y_{1}^{2}} d x

When to use. When one solution is known for a second order linear DE.

Trap. DE must be in standard form $y^{''} + P (x) y^{'} + Q (x) y = 0$ .

Wronskian and Linear Independence

W (y_{1}, y_{2}) = ∣ \begin{array}{cc} y_{1} & y_{2} \\ y_{1}^{'} & y_{2}^{'} \end{array} ∣

When to use. Checking linear independence of solutions.

Trap. Wronskian can be zero at some points but not identically zero for independence.

Solving DEs by Inspection (Special Forms)

d (x y) = x d y + y d x

d (\frac{y}{x}) = \frac{x d y - y d x}{x^{2}}

d (\frac{x}{y}) = \frac{y d x - x d y}{y^{2}}

When to use. When DE can be rearranged into exact differential forms.

Trap. Group terms to match exact differentials.

Applications: Growth and Decay Problems

N (t) = N_{0} e^{k t}

When to use. Problems involving population growth, radioactive decay, Newton's law of cooling.

Trap. Identify whether k is positive (growth) or negative (decay).

Applications: Newton's Law of Cooling

T (t) = T_{s} + (T_{0} - T_{s}) e^{- k t}

When to use. Problems where an object cools or heats to ambient temperature.

Trap. Ensure $T_{s}$ is the surrounding constant temperature.

Read the full chapter: Differential Equations for JEE, with what JEE asks and the common traps.