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 |