Dr. Brajesh Kumar Jha

First Order Homogeneous Differential Equations (Solved Examples) A differential equation of the form \[ \frac{dy}{dx}=f\left(\frac{y}{x}\right) \] is called a homogeneous differential equation . We use the substitution \[ y=vx \quad \text{or} \quad v=\frac{y}{x} \] so that \[ \frac{dy}{dx}=v+x\frac{dv}{dx} \] Example 1: Solve \( \frac{dy}{dx}=\frac{x+y}{x} \) View Solution \[ \frac{dy}{dx}=1+\frac{y}{x} \] Let \[ v=\frac{y}{x}, \quad y=vx \] \[ \frac{dy}{dx}=v+x\frac{dv}{dx} \] Substitute \[ v+x\frac{dv}{dx}=1+v \] \[ x\frac{dv}{dx}=1 \] \[ \frac{dv}{dx}=\frac{1}{x} \] Integrate \[ v=\ln|x|+C \] Since \(v=\frac{y}{x}\) \[ \frac{y}{x}=\ln|x|+C \] \[ y=x(\ln|x|+C) \] Example 2: Solve \( \frac{dy}{dx}=\frac{x+y}{x-y} \) View Solution Let \[ v=\frac{y}{x}, \quad y=vx \] \[ \frac{dy}{dx}=v+x\frac{dv}{dx} \] Substitute \[ v+x\frac{dv}{dx}=\frac{1+v}{1-v} \] \[ x\frac{dv}{dx}=\frac{1+v}{1-v}-v \] \[ x\frac{dv}{dx}=\frac{1+v^2}{...

First Order Differential Equations using Separation of Variables Below are solved examples of first order differential equations using the separation of variables method . Example 1: Solve \( \frac{dy}{dx}=3x^2 \) View Solution Separate variables \[ dy = 3x^2 dx \] Integrate both sides \[ \int dy = \int 3x^2 dx \] \[ y = x^3 + C \] Example 2: Solve \( \frac{dy}{dx}=xy \) View Solution Separate variables \[ \frac{dy}{y}=x\,dx \] Integrate \[ \int \frac{1}{y}dy = \int x\,dx \] \[ \ln |y| = \frac{x^2}{2}+C \] \[ y = Ce^{x^2/2} \] Example 3: Solve \( \frac{dy}{dx}=\frac{x}{y} \) View Solution Separate variables \[ y\,dy = x\,dx \] Integrate \[ \int y\,dy = \int x\,dx \] \[ \frac{y^2}{2} = \frac{x^2}{2} + C \] \[ y^2 = x^2 + C \] Example 4: Solve \( \frac{dy}{dx}=y^2 \) View Solution Separate variables \[ \frac{dy}{y^2}=dx \] Integrate \[ \int y^{-2}dy = \int dx \] \[ -\frac{1}{y}=x+C \] \[ y=\frac{1}{C-x} ...

First Order Differential Equations using Separation of Variables Advanced Application Problems (Separation of Variables) The following examples illustrate applications of first order differential equations in population growth, cooling law, chemical reactions, and physics. Example 11 (Population Growth): If population grows according to \( \frac{dP}{dt}=kP \), find \(P(t)\). View Solution Separate variables \[ \frac{dP}{P}=k\,dt \] Integrate \[ \int \frac{dP}{P}=\int k\,dt \] \[ \ln P = kt + C \] \[ P = Ce^{kt} \] Example 12 (Newton's Law of Cooling): \[ \frac{dT}{dt}=-k(T-T_s) \] Find \(T(t)\). View Solution Separate variables \[ \frac{dT}{T-T_s}=-k\,dt \] Integrate \[ \ln |T-T_s|=-kt+C \] \[ T-T_s=Ce^{-kt} \] \[ T=T_s+Ce^{-kt} \] Example 13 (Radioactive Decay): Solve \( \frac{dN}{dt}=-kN \) View Solution \[ \frac{dN}{N}=-k\,dt \] Integrate \[ \ln N=-kt+C \] \[ N=Ce^{-kt} \] Example 14 (Chemical Reaction): So...