A differential equation is the mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering Differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics> now let us see about the ordinary differential equation.

Problems in Ordinary Differential Equations

Solution to the Example in ordinary differential equation 1

Find a general solution to the differential equation.

2y' = sin(4x)

Write the differential equation of the form y ' = f(x).

y'= (1/2) sin(4x)

Integrate both sides

y'dx = (1/2) sin(4x) dx

Let u = 2x so that du = 2 dx, the right side becomes

y =(1/2) sin(2u) du

Which gives.

y=(-2/2) cos(2u) = (-1) cos (4x)

ind a general solution to the differential equation.

y'e^-x + e^3x = 0

Solution to Example in ordinary differential equation 2:

Multiply all terms of the equation by e x

y ' = - e^4x

Integrate both sides of the equation

y ' dx = - e^4x dx

Let u = 4x so that du = 4 dx, write right side of the term of u

Solution to the Example in ordinary differential equation 1

Find a general solution to the differential equation.

2 y ' = sin(4x)

Write the differential equation of the form y ' = f(x).

y ' = (1/2) sin(4x)

Integrate both sides

y ' dx = (1/2) sin(4x) dx

Let u = 2x so that du = 2 dx, the right side becomes

y = (1/2) sin(2u) du

Which gives.

y = (-2/2) cos(2u) = (-1) cos (4x)

ind a general solution to the differential equation.

y 'e^-x + e^3x = 0

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Solution to Example in ordinary differential equation 2:

Multiply all terms of the equation by e x

y ' = - e^4x

Integrate both sides of the equation

y'dx = - e^4x dx

Let u = 4x so that du = 4 dx, write right side of the term of u

y = (-1/4) e u du

Which gives.

y = (-1/4) e u = (-1/4) e^4x

Solve the general solution for differential equation

dy / dx + y / x = - 3 for x > 0

Solution to Example in ordinary differential equation 3:

We first find P(x) and Q(x)

P(x) = 1 / x and Q(x) = - 3

The integrating factor u(x) is given by

u(x) = e P(x) dx

= e^(1 / x) dx

= e^ln|x| = | x | = x since x > 0.

now substitute the u(x)= x and Q(x) = - 3 in the equation u(x) y = u(x) Q(x) dx to get

x y = -3 xdx

Integrate the right hand term to get

xy = -x^3 + C , C is a constant of integration.

Solve the above for y to get

y = C / x x^2 Which gives.

y = (-1/4)e^u = (-1/4)e^4x

Solve the general solution for differential equation

dy / dx + y / x = - 3 for x > 0

Solution to Example in ordinary differential equation 3:

We first find P(x) and Q(x)

P(x) = 1 / x and Q(x) = - 3

The integrating factor u(x) is given by

u(x) = e^P(x) dx

= e^(1 / x) dx

= e^ln|x| = | x | = x since x > 0.

now substitute the u(x)= x and Q(x) = - 3 in the equation u(x) y = u(x) Q(x) dx to get

x y = -3 xdx

Integrate the right hand term to get

x y = -x3 + C , C is a constant of integration.

Solve the above for y to get

y = C / x x^2

Examples for Ordinary Differential Equation

Example in ordinary differential equation 4: X^4-3x^2+14x+5.

Sol : dy/dx=4x^3-6x+14.

Example in ordinary differential equation 5:

Differentiate the given equation:

4x^5+3x^3+5.

Sol : dy/dx= 20x^4+9x^2.