Math Notes

Differential Equation

First-order & higher-order ODEs—separable, homogeneous, exact, linear—with real-life applications.

Solved Problems

Topics
Separable Ordinary Differential Equation
Obtain the general equation of \( \tan^2 y dy = \sin^3 x dx\)Obtain the general equation of \( (1+\ln x)dx + (1 + \ln y)dy\)Find the general solution of the equation \( xy^3 dx +e^{x^2}dy =0\)Obtain the general solution of \( \sin x \sin y dx + \cos x \cos y dy = 0 \)Obtain the general solution of \((1-x)y' = y^2\)
Exact Equations
Solve the equation \( (x + 2y)dx + (2x +y ) dy = 0\)
Homogeneous Differential Equation
Obtain a family of solutions \( (x - 2y)dx + (2x +y )dy = 0\)
Elementary Applications
A thermometer reading is \(18^oF\) is brought into a room where the temperature is \(70^oF\); 1 min later the thermometer reading is \(31^oF\). Determine the temperature reading as a function of time and, in particular, find the temperature reading 5 min after the thermometer is first brought into the room.
Tags:
differential equations ODE separable equations homogeneous exact equations linear ODE general solution particular solution applications