Math · Calculus

Differential Equations formulas for JEE

Every Differential Equations formula you need for JEE, grouped by concept.

23 formulas4 concepts

01Differential Equations Foundation 02Separable and Homogeneous DE 03Linear Differential Equations 04Methods of Solving Differential Equations

Differential Equations Foundation

3 formulas

Orthogonal Trajectories Substitution

\frac{dy}{dx} \to -\frac{dx}{dy}

Replacement rule to find the differential equation of an orthogonal family of curves.

applies whenApplied to the differential equation representing the primary family of curves.

differential-equationsorthogonal-trajectoriesjee-advanced

Length of Subnormal

L_{SN} = \left| y \frac{dy}{dx} \right|

Formula for the length of the subnormal on the x-axis for a curve y=f(x).

applications-of-derivativesgeometryjee-advanced

Length of Subtangent

L_{ST} = \left| y \frac{dx}{dy} \right|

Formula for the length of the subtangent on the x-axis for a curve y=f(x).

applications-of-derivativesgeometryjee-advanced

Separable and Homogeneous DE

5 formulas

Continuous Growth/Decay

P(t) = P_0 e^{kt}

Solution to proportional growth rate DEs like compound interest or bacteria population.

applies whenDerived from

dP/dt = kP

differential-equationsapplications

Homogeneous Function Condition

F(\lambda x, \lambda y) = \lambda^n F(x, y)

Condition for a function to be homogeneous of degree n.

applies whenFor any non-zero constant

\lambda

differential-equationshomogeneous

Homogeneous Substitution (y=vx)

\frac{dy}{dx} = v + x\frac{dv}{dx}

Substitution to reduce a homogeneous differential equation to variable separable form.

applies whenUsed when

dy/dx = F(x,y)

where F is a homogeneous function of degree 0.

differential-equationshomogeneous

Homogeneous Substitution (x=vy)

\frac{dx}{dy} = v + y\frac{dv}{dy}

Alternative substitution for homogeneous differential equations.

applies whenUsed when

dx/dy = F(x,y)

where F is a homogeneous function of degree 0.

differential-equationshomogeneous

Separation of Variables

\int \frac{1}{h(y)} dy = \int g(x) dx + C

General solution form for a separable first-order differential equation.

applies when

h(y) \neq 0

differential-equationsseparable

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Linear Differential Equations

4 formulas

Integrating Factor (dy/dx)

I.F. = e^{\int P dx}

Integrating factor for a standard linear differential equation.

applies whenFor DE in form

dy/dx + Py = Q

, where P is a function of x only.

differential-equationslinearintegrating-factor

Integrating Factor (dx/dy)

I.F. = e^{\int P_1 dy}

Integrating factor for the alternative linear differential equation form.

applies whenFor DE in form

dx/dy + P_1x = Q_1

, where P1 is a function of y only.

differential-equationslinearintegrating-factor

Linear First-Order Solution (dy/dx)

y \cdot (I.F.) = \int Q \cdot (I.F.) dx + C

General solution of a standard linear differential equation.

applies whenFor DE in form

dy/dx + Py = Q

differential-equationslinear

Linear First-Order Solution (dx/dy)

x \cdot (I.F.) = \int Q_1 \cdot (I.F.) dy + C

General solution of the alternative linear differential equation form.

applies whenFor DE in form

dx/dy + P_1x = Q_1

differential-equationslinear

Methods of Solving Differential Equations

11 formulas

Bernoulli's Equation Reducible to Linear

y^{-n}\frac{dy}{dx} + Py^{1-n} = Q \implies t = y^{1-n}

Substitution to reduce Bernoulli's differential equation into a linear form.

applies whenFor DE in form

dy/dx + Py = Qy^n

differential-equationsbernoullijee-advanced

Exact Differential Equation Condition

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

Necessary and sufficient condition for a first order DE to be exact.

applies whenFor DE in form

M(x,y)dx + N(x,y)dy = 0

differential-equationsexactjee-advanced

Exact Differential of Logarithmic Radius

d\left(\frac{1}{2}\ln(x^2+y^2)\right) = \frac{x dx + y dy}{x^2 + y^2}

Standard exact differential identity for the natural log of distance to origin.

differential-equationsexact-differentialsjee-advanced

Exact DE Solution Form

\int_{y=\text{const}} M dx + \int N_{\text{free of } x} dy = C

Direct integration method for solving exact differential equations.

applies whenMust satisfy the exactness condition.

differential-equationsexactjee-advanced

Exact Differential of Arctan

d\left(\tan^{-1}\frac{y}{x}\right) = \frac{x dy - y dx}{x^2 + y^2}

Standard exact differential identity involving arctangent of y/x.

differential-equationsexact-differentialsjee-advanced

Exact Differential of Quotient x/y

d\left(\frac{x}{y}\right) = \frac{y dx - x dy}{y^2}

Standard exact differential identity for the quotient x/y.

applies when

y \neq 0

differential-equationsexact-differentialsjee-advanced

Exact Differential of Product

d(xy) = x dy + y dx

Standard exact differential identity for the product of x and y.

differential-equationsexact-differentialsjee-advanced

Exact Differential of Quotient y/x

d\left(\frac{y}{x}\right) = \frac{x dy - y dx}{x^2}

Standard exact differential identity for the quotient y/x.

applies when

x \neq 0

differential-equationsexact-differentialsjee-advanced

Reducible to Homogeneous (Intersecting)

x = X + h, y = Y + k

Shift of origin substitution to make linear fractional DEs homogeneous.

applies whenFor

dy/dx = (a_1x+b_1y+c_1)/(a_2x+b_2y+c_2)

where

a_1/a_2 \neq b_1/b_2

differential-equationssubstitutionjee-advanced

Reducible to Homogeneous (Parallel)

t = a_1x + b_1y

Substitution for linear fractional DEs with parallel line coefficients.

applies whenFor

dy/dx = (a_1x+b_1y+c_1)/(a_2x+b_2y+c_2)

where

a_1/a_2 = b_1/b_2

differential-equationssubstitutionjee-advanced

Reducible to Variable Separable

t = ax+by+c \implies \frac{dt}{dx} = a + b f(t)

Substitution for DEs featuring a linear combination of variables.

applies whenFor DE in form

dy/dx = f(ax+by+c)

differential-equationssubstitutionjee-advanced

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