First-Order ODEJEE Main

Concept

A differential equation links a function to its own rate of change. The separable equation $\frac{dy}{dx} = ky$ describes anything growing or decaying in proportion to itself — its solution is the exponential $y = y_0 e^{kx}$. The slope field shows the direction every solution must follow.

Key formula

Derivation

Separate the variables: $\frac{dy}{y} = k\,dx$. Integrating both sides gives $\ln|y| = kx + C$.

Exponentiating, $y = Ae^{kx}$, and applying $y(0)=y_0$ fixes $A=y_0$. The constant $k$ sets the doubling time ($\ln 2/k$ for growth) or half-life ($\ln 2/|k|$ for decay).

Scenarios to explore

• First-Order ODE — Slope field and exponential solution of dy/dx = ky.

Real-world applications

• Radioactive decay and population growth.
• Newton's law of cooling.
• RC and RL circuit charging/discharging.

JEE exam tips

• Always separate variables first for $dy/dx = f(x)g(y)$ equations.
• Half-life $= \ln 2/|k|$; the solution never actually reaches zero.
• A negative $k$ flips growth into decay; $k=0$ gives a constant solution.

Common mistakes

• Dropping the constant of integration $C$ before applying the initial condition.
• Forgetting the modulus when integrating $1/y$.
• Confusing the rate $k$ with the solution value.

Exam traps to avoid

• $e^{kx}$ is never zero — exponential decay approaches but never hits the axis.
• The slope field's segments depend on $y$ only here, so they are horizontal where $y=0$.