# Solve Simple Differential Equations

This is a tutorial on solving simple first order differential equations of the form

y ' = f(x)
A set of examples with detailed solutions is presented and a set of exercises is presented after the tutorials. Depending on f(x), these equations may be solved analytically by integration. In what follows C is a constant of integration and can take any constant value.

 Example 1: Solve and find a general solution to the differential equation. y ' = 2x + 1 Solution to Example 1: Integrate both sides of the equation. ò y ' dx = ò (2x + 1) dx which gives y = x 2 + x + C. As a practice, verify that the solution obtained satisfy the differential equation given above. Example 2: Solve and find a general solution to the differential equation. 2 y ' = sin(2x) Solution to Example 2: Write the differential equation of the form y ' = f(x). y ' = (1/2) sin(2x) Integrate both sides ò y ' dx = ò (1/2) sin(2x) dx Let u = 2x so that du = 2 dx, the right side becomes y = ò (1/4) sin(u) du Which gives. y = (-1/4) cos(u) = (-1/4) cos (2x) Example 3: Solve and find a general solution to the differential equation. y 'e -x + e 2x = 0 Solution to Example 3: Multiply all terms of the equation by e x and write the differential equation of the form y ' = f(x). y ' = - e 3x Integrate both sides of the equation ò y ' dx = ò - e 3x dx Let u = 3x so that du = 3 dx, write the right side in terms of u y = ò (-1/3) e u du Which gives. y = (-1/3) e u = (-1/3) e 3x Exercises: Solve the following differential equations. a) 2y ' = 6x b) y ' cos x = sin(2x) c) y ' e x = e 3x Solutions to the above exercises a) y = (3/2) x 2 + C b) y = -2 cos x + C c) y =(1 / 2) e 2x + C More references on Differential Equations Differential Equations - Runge Kutta Method