JEE Maths›Differential Equations›Formulas
Formula Sheet

Differential Equations Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

Order and Degree of a Differential Equation

#1
Order=highest order derivative,Degree=power of highest order derivative (polynomial form)\text{Order} = \text{highest order derivative}, \quad \text{Degree} = \text{power of highest order derivative (polynomial form)}

💡 Order is always defined. Degree is defined only when the DE is a polynomial in its derivatives. If sin(y') or e^(y'') appears, degree is not defined.

Variable Separable Form

#2
dydx=f(x) g(y)  ⟹  ∫dyg(y)=∫f(x) dx+C\frac{dy}{dx} = f(x)\,g(y) \implies \int \frac{dy}{g(y)} = \int f(x)\,dx + C

💡 Separate all y terms (with dy) to one side and all x terms (with dx) to the other, then integrate both sides.

Homogeneous Differential Equation

#3
dydx=F ⁣(yx),Put y=vx  ⟹  v+xdvdx=F(v)\frac{dy}{dx} = F\!\left(\frac{y}{x}\right), \quad \text{Put } y = vx \implies v + x\frac{dv}{dx} = F(v)

💡 After solving for v, substitute back v = y/x to get the solution in terms of x and y.

Linear First-Order DE

#4
dydx+P(x) y=Q(x),IF=e∫P dx,y⋅IF=∫Q⋅IF dx+C\frac{dy}{dx} + P(x)\,y = Q(x), \quad \text{IF} = e^{\int P\,dx}, \quad y \cdot \text{IF} = \int Q \cdot \text{IF}\,dx + C

💡 First rewrite the DE in standard linear form. The integrating factor (IF) multiplies both sides. Remember: the formula also works as dx/dy + P(y)x = Q(y).

Bernoulli's Equation

#5
dydx+P(x) y=Q(x) yn,Divide by yn, put v=y1−n\frac{dy}{dx} + P(x)\,y = Q(x)\,y^n, \quad \text{Divide by } y^n, \text{ put } v = y^{1-n}

💡 After substituting v = y^(1-n), the equation reduces to a linear DE in v. Solve that linear DE, then convert back to y.

Exact Differential Equation

#6
M dx+N dy=0 is exact if ∂M∂y=∂N∂xM\,dx + N\,dy = 0 \text{ is exact if } \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

💡 Solution: integrate M w.r.t. x (treating y as constant), then add terms from N that are not already present. The solution is F(x,y) = C.

Formation of DE by Eliminating Constants

#7
Family with n arbitrary constants  ⟹  differentiate n times and eliminate all constants\text{Family with } n \text{ arbitrary constants} \implies \text{differentiate } n \text{ times and eliminate all constants}

💡 The order of the resulting DE equals the number of arbitrary constants in the family of curves.

Orthogonal Trajectories

#8
Replace dydx with −dxdy in the DE of the given family\text{Replace } \frac{dy}{dx} \text{ with } -\frac{dx}{dy} \text{ in the DE of the given family}

💡 Orthogonal trajectories cut the given family at right angles. After replacement, solve the new DE to get the trajectory equation.

Exponential Growth and Decay

#9
dNdt=kN  ⟹  N=N0 ekt\frac{dN}{dt} = kN \implies N = N_0\,e^{kt}

💡 k > 0 for growth, k < 0 for decay. Half-life: t_{1/2} = log(2)/|k|. Use log, not ln, per JEE convention.

Newton's Law of Cooling

#10
dTdt=−k(T−Ts),T=Ts+(T0−Ts) e−kt\frac{dT}{dt} = -k(T - T_s), \quad T = T_s + (T_0 - T_s)\,e^{-kt}

💡 T_s is the surrounding temperature, T_0 is the initial temperature of the body, and k > 0. The body cools exponentially toward T_s.

←
Previous
Application of Derivatives
Next
Conic Sections
→