💻 Second order differential

This lesson covers core applied mathematics concepts and their agricultural applications for BSc Agriculture learners.

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MATHS :: Lecture 20 :: Second order differential equations

Second order differential equations with constant coefficients

The general form of linear Second order differential equations with constant coefficients is (aD2 + bD + c ) y = X (i) Where a,b,c are constants and X is a function of x.and D = When X is equal to zero, then the equation is said to be homogeneous. Let D = m Then equation (i) becomes am2 +bm +c = 0 This is known as auxiliary equation. This quadratic equation has two roots say m1 and m2. The solution consists of one part namely complementary function (ie) y = complementary function

[Linear Second-order Equations- Fundamentals](<matvdo/sec-diffequ/Linear Second-order Equations- Fundamentals.mov>)

Second order DEs1

Second order DEs2

Second order DEs3

Complementary Function

Case (i)

If the roots (m1 & m2) are real and distinct ,then the solution is given by where A and B are the two arbitrary constants.

Case (ii)

If the roots are equal say m1 = m2 = m, then the solution is given by where A and B are the two arbitrary constants.

Case (iii)

If the roots are imaginary say and Where and are real. The solution is given by where A and B are arbitrary constants.

Particular integral

The equation (aD2 + bD + c )y = X is called a non homogeneous second order linear equation with constant coefficients. Its solution consists of two terms complementary function and particular Integral. (ie) y = complementary function + particular Integral Let the given equation is f(D) y(x) = X y(x) =

Case (i)

Let X= and f() Then P.I = =

Case (ii)

Let X = P(x) where P(x) is a polynomial Then P.I = P(x) = [f(D)]-1 P(x) Write [f(D)]-1 in the form (1 (1 and proceed to find higher order derivatives depending on the degree of the polynomial.

Newton's Law of Cooling

Rate of change in the temperature of an object is proportional to the difference between the temperature of the object and the temperature of an environment. This is known as Newton's law of cooling. Thus, if is the temperature of the object at time t , then we have

-k() This is a first order linear differential equation.

Population Growth

The differential equation describing exponential growth is This equation is called the law of growth, and the quantity K in this equation is sometimes known as the Malthusian parameter.

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Summary Cheat Sheet

• Focus on core formulas, definitions, and solved patterns from this lesson.
• Practice stepwise derivations and numerical substitutions carefully.
• Connect each concept to practical agricultural problem-solving contexts.

References

1 source

Primary classroom notes and standard BSc Agriculture applied mathematics references.