9 Concepts to Thoroughly Understand Before Starting Your Differential Equations Assignment

In calculus, you learn about differential equations and how to solve them. Finding solutions to differential equations is applied in different disciplines, including engineering, economics, biology, and mechanics. We have many classes of differential equations that you can solve using simple techniques such as integration and substitution and complex ones as the course advances.

Before you start working on your differential equations assignment, you should master the basic concepts and solve many similar problems. You can get practice questions on different subtopics from your college library and other online learning materials. If you miss a concept in class and experience difficulty understanding it on your own, get help from an experienced online tutor here.

To make your study and preparation for differential equations assignment easier, we have analyzed the entire course and highlighted the crucial concepts where students usually are tested. Read on and find out 9 concepts you must understand before starting your assignment.

Basic Definitions: Differential Equations, Order, and Degree

A mathematical equation containing an unknown function and its derivative is a differential equation. The equation yʹ =2x is a differential equation with an unknown function y and a derivative. The above equation can be re-written as dy/dx = 2x. 2x is the derivative of y with respect to x, while x is the independent variable.

The highest order derivative in a differential equation is its order. For example, d2x/dy2 =3x is a second order derivative. The highest power to which a derivative is raised in a differential equation is its degree. In the equation (d2x/dy2)3 +dy/dx=3x, the order is two, while the degree is three.

Other important terms in this course include:

• A solution to a differential equation is a function y= f(x) that satisfies the DE when the derivatives are substituted in the equation. Answers can be general or particular depending on the nature and order of the equation. For example, y=mx+c has constants m and c. A particular solution takes the form y= f(x) and has no arbitrary constants.
• The initial value problem is a differential equation with initial conditions, for example, dy/dx=5x+2, y=0 is an IVP.
• Linear and nonlinear equations. A linear equation has all functions combined add up to zero, that is, dy/dx+p(x) = 0. A non-linear differential equation has an unknown function that is not linear, for example, dy/dx + p(t2) = 2. The unknown function, t is not linear; therefore, the equation is nonlinear.
• Ordinary and partial differential equations. An ordinary differential equation (ODE) contains one or more functions of one independent variable, while a partial differential equation (PDE) contains two or more independent variables.

First Order Differential Equations and How to Solve Them

Equations containing derivatives of order 1 are called first-order differential equations. The equation dy/dx = 2x is an example of a first-order differential equation because the highest power of the derivative is one.

First-order differential equations can be linear, homogeneous, non-homogeneous, separable, or exact. The method you choose depends on lineality, homogeneity, and the number of variables. For example, solving a single variable DE involves integrating the derivative and using the initial conditions to find the constant. Here are the common methods you can use to solve these equations.

• Substitution

  Consider this equation: dy/dx+p(x)y = q(x) where p and q are functions of x. The equation can be re-written as dy/dx= p(x)y + q(x)

  Let the two functions of x be denoted as u and v. the derivative becomes dy/dx = udv/dx+ vdu/dx. This is followed by separating the variables and integrating them before substituting them with the original equation.

• Separation of variables: You can solve the first-order DE with two or more variables by making them separable and finding their integrals.

  Example: dy/dx +y = -x2 is solvable by separating variables as dy/y = - x2dx and integrating them with respect to x and y respectively.

• Solving by variation of constants. This method is applied when solving non-homogeneous linear differential equations where you replace a function with a constant of integration before solving.
• Substitution on Bernoulli equation. If your equation takes the polynomial form, you can solve it by substitution before determining the roots.

Second Order Differential Equations

A differential equation whose highest order derivative is 2 is a second order DE. For example, d2x/dy2 =3x is second order equation. A differential equation of order 2 can be homogeneous or non-homogeneous. There are several techniques you can use in solving these equations, such as:

• Variation of parameters. If you are solving a non-homogeneous second-order DE, this method is ideal. You can obtain the general solution by substituting the constant and finding roots for the polynomial.
• Use of undetermined coefficients. Here, you guess the coefficients to the function and solve the equation without determining the coefficients first. Consider the equation y″ + y′ + 2y = x2. You can write the equation as ay″ + by′ + c2y = x2, and assuming x2 is 0, solve the polynomial without determining the values of a,b and c first.
• Using the Wronskian. Independent solutions to second-order DEs form a matrix with linearly independent elements whose determinant is not zero. The Wronskian will help you determine whether a set of answers to your equation are applicable or not.

For homogeneous second-order differential equations, you can use these methods to work out their general and particular solutions. Some of the ways you can use to solve the equations are:

• Superposition method where you derive a characteristic polynomial from the functions and work out its roots.
• If your equation has double or repeated roots, you can use the reduction of order method and find the solutions to your DE.
• After obtaining a fundamental set of equations by superposition, you can use the Wronskian to test for their validity.

Higher Order Differential Equations

Higher order differential equations have an order derivative of 3 or more. If conversant with methods used in second-order derivatives, you are set for these equations. The techniques in order 2 are applicable in higher orders, and the workings are similar.

• If the Wronskian is non-zero, your equation has linearly independent functions. This implies that you can get the general and the particular solution to the equation by reduction of orders.
• If the equation is non-homogeneous, use the method of undetermined coefficients to solve the polynomial formed by the functions.
• Form a matrix and solve it by row reduction or zeroing method if non-homogeneous and inverse method if homogeneous.
• Use the Euler equation to find solutions to a power series. If the combination of functions in the DE adds up to pn, you can apply the Euler equation.
• Work out simple solutions by finding integrals to the derivatives

Partial Differential Equations

A differential equation with multiple unknown variables and their derivatives is called a partial differential equation (PDE). After you identify an equation as a PDE, you can solve it by splitting it into multiple ordinary equations. Note that PDEs occur in different orders, and solving them requires you to apply the methods discussed above.

You can solve a PDE using two or more techniques depending on the ODEs you obtain after splitting it. Some common methods include;

• Making variables separable and working out their integrals
• Adding undetermined coefficients to the polynomial before solving it to obtain roots. You can test for the roots using the Wronskian.
• Using a Laplace equation to work out independent solutions to the equation. You can also use the Laplace transform table to test for solutions.
• Use the substitution method to work out solutions to linear PDEs.

Solving PDEs is not limited to the above techniques but involves most of the theorems and formulae in the course. It is, however, crucial that you identify the type of equations and convert them correctly, and you can do this through consistency and practice.

Systems of Differential Equations

A group of differential equations makes a system. To find a solution to a DE system of DEs, you should solve them simultaneously using the basic algebraic methods. For example, you can form a matrix by listing all the coefficients to the functions in rows and columns and find the solution to the matrix formed.

The common algebraic methods for solving systems of linear equations are the inverse of the matrix and row reduction method. However, you should note that these methods only apply to linear equations. A good grasp of concepts in matrices is a prerequisite to this course, therefore, you can review your notes on linear algebra.

Before you start solving equations, ensure you test whether they are linearly independent. Basic concepts in vectors and mechanics are essential to identify eigenvalues and eigenvectors. Note the following:

• Eigenvalues are scalar values obtained from the latent roots of a linear equation. You should use them to obtain the Eigenvectors for the system, and when you use them, ensure you maintain the direction of the system.
• If your equations have real roots, the Eigenvectors you obtain are classified as real. If you get double roots, the vectors are repeated.
• When working on a system of non-homogeneous equations, use methods for solving non-homogeneous equations such as row reduction and variation of parameters.
• Laplace transform will help you work out systems of equations by obtaining step functions and solving initial value problems.
• The inverse to the Laplace transform helps work out the convolution integral to your system equation to get the initial value problem solution.

The approach to any system of equations is all you have learned in the various sections of the course. If you understand how to solve first order, second order, and higher order equations, you can apply any suitable method and your knowledge in linear algebra to solve any problem.

Laplace Transform and How to Use It

Denoted as L{f(t)}=F(s), the Laplace transform helps you convert differential equations to algebraic equations before solving them. The Laplace transform works best when solving linear differential equations, but you cannot use it on non-linear equations.

You can solve linear differential equations using the Laplace transform in the following four steps:

1. Use the derivative property of the Laplace transform to convert the variables in your equation. For example, given the differential equation y(t) + 3y(t) = f(t), its transform becomes;

                  y(t)-y(t-1) given the initial conditions y(0)=-2

2. Put the initial conditions into the resulting equation. Suppose the initial condition in the above equation is f(t) = 1/s - e-s1/s, the equation becomes sy(s)+2+3y(s)= 1/s - e-s1/s
3. Solve for the unknown, y(s). In the above equation, y(s) becomes 1/s(s+2) - e-s 1/s(s+2) - 3 1/(s+2).
4. After the third step, you can get the result from the Laplace transform table. In some cases, the results may not be on the table, therefore, you will need to use the inverse Laplace transform to obtain the results.

The Laplace transform is widely used in first and second-order equations and systems of equations. You can obtain the convolution integral in systems using the L. transform table. It is challenging if a system has zero initial conditions.

Series Solutions to Differential Equations

You can work out some equations by finding a series for their solutions. This requires prior knowledge of the Taylor series learnt in the introduction to calculus. This series is limited to a given degree's polynomial functions, say n. Before applying the Taylor and other series to obtain power solutions, you should study and master the features of a power series, including:

• The ratio test will help you determine whether an equation will converge or diverge. Different operations on the series, including addition, subtraction, and differentiation, are crucial at this point. In addition, you should learn how to determine the index shifts in a power series.
• In the Taylor series, two functions are essential, ex and cosx. You should construct the T.series for the two functions and show how to use them in any polynomial.
• Two important points when working on series solutions are ordinary and singular points. Consider the following polynomial:

        yʺ+P(x)y′ + Q(x)y = 0

Given the limit x=x0, if the functions P(x) and Q(x) are differentiable as x tends to x0 they are ordinary. This implies that within the range x→ x0, the functions are ordinary. Any point which is not ordinary (or is non-differentiable within the given limits) is singular.

In addition to the Taylor series, you can solve functions with sine and cosine functions using the Fourier series. Two series, the cosine and sine series, make the Fourier series denoted as

∑n=0∞ Ancos(n𝝅x/l) +∑Bnsin (n𝝅x/l). You can apply this series on order 2 derivatives with periodic functions.

Note that the sine and cosine series are suitable when your equation has only one periodic function. The sine series takes ∑Bnsin (n𝝅x/l), while the cosine series is given as ∑n=0∞ Ancos(n𝝅x/l).

Boundary Value Problems and the Fourier Series

A boundary value problem has solutions and derivatives specified within a range or at different points. For example, a second order DE given as yʺ +3y = 0, with y(0)= 0 and y(𝛑/4)= 6 has its solutions between 0 and 4 for the specified points. Some boundary value problems are periodic and require you to solve them using the F. series.

Also, you can solve orthogonal functions using the F.series by calculating the integral value of the weight function. The functions are orthogonal if you obtain zero as the product of the integral and the weighted value.

Final Thoughts

Solving your differential equations assignment can be easy if you work diligently on these nine areas and solve more problems from each section. Many techniques are highlighted above, and some are specific to given subtopics. Identifying a suitable method for solving any differential equation from the many highlighted requires deep content mastery.

How do you master all these techniques and apply them correctly? Well, it is simple. Dedicate each day of your study to a specific area and work on it exhaustively. Start by definitions and deriving equations, and solve more relevant questions. In addition, seek clarification from your peers for any challenging problem.

While solving your assignment questions, read them well and find a suitable solution method. This wil help you work out the problem systematically, showing all the required details and steps.