Cauchy’s and Euler’s Equations

Cauchy and Euler’s equations are helpful to solve the problems of mathematics. One equation is generally used to solve the mathematical problems that come from the differential equation. The learners and the instructors are able to solve the problems of ordinary differential equations by using another method; the difficulties have been reduced by the use of this equation. Furthermore, this equation is often known as simply Euler’s equation that states the agricultural equation effectively. In the contemporary world, not only the practitioners but also researchers use this to solve various problems of ordinary differential equations that have been coming into the calculation part.

Definition of Cauchy’s and Euler’s   equations

The differential equation, Cn Pn Q(n)(P) + Cn – 1 Pn – 1 Q(n – 1)(P) + Cn-2 Pn-2 Q(n-2)(P) +———— + C0 Q (P) = 0, where Q is the dependent variable of the P , P is independent variable, Cn where n = 0, 1, 2, …..  is the constant, and Q(n)(P) represents the derivative of the function Q (P). The mentioned differential equation of order N is known as the Cauchy’s and Euler’s equations. The mentioned equation is helpful in the theory of the linear differential equation. The mentioned equation has direct use in the Fourier methods and this is a reason it is important for the theory of linear differential equations. The Cauchy third order equation can be derived from the above equation and that is C3 P3 Q(3)(P) + C2 P2 Q(2)(P) + C1 P1 Q(1)(P) + C0 Q (P) = 0, where Q is a dependent variable, Q(3)(P) 3rd derivative of the function Q (P)  that is a dependent variable and depends on the value of P, Q(2)(P) is 2nd derivative of the function Q (P) and Q(1)(P) is 1st derivative of the function Q(P). Similarly, the equation, C2 P2 Q(2)(P) + C1 P1 Q(1)(P) + C0 Q (P) = 0 is second orders Cauchy’s and Euler’s equations. Q(2)(P) is the 2nd derivative of the function Q (P), Q(1)(P) is the 1st derivative of the function Q(P), Q(P) is a dependent variable that depends on the value of the P; and C0, C1, and C2 are constants.

Uses of Cauchy’s and Euler’s Equations in Differential Equation

The mentioned equation is an important formula in the context of differential equation formulation because these can be employed directly in Fourier’s series for solving partial differential equations in mathematics. In order to solve other Fourier’s series equations, they can directly use second-degree Cauchy’s Euler’s equations such as ap2qn-2+zy=0. In this formula, z works as an indefinite constant that is produced due to partial differential equation insolvency. Furthermore, this is an account for use in applied literature study techniques. A second contradiction about the second-order degree formula is this can be used only for theoretical problem solutions. The second-order formula of Cauchy’s Euler’s equation is a single formula for example solution of differential equation with assisting non–constant coefficients that is recognized for close form solution. Furthermore, this also can be solved from the change of variables (p, q) into another coefficient such as (t, z). This change of arbitrarily can be given by p=et, q(x) =z (t). 

Problems of Differential Equation

In order to solve the problems of differential equations, the learners or the students have to put Q = Pr in the given differential equation of the form Cn Pn Q(n)(P) + Cn – 1 Pn – 1 Q(n – 1)(P) + Cn-2 Pn-2 Q(n-2)(P) +———— + C0 Q (P) = 0, where Q is the dependent variable of the P, P is the independent variable, Cn where n = 0, 1, 2, …..  is the constant, and Q(n)(P) represents the derivative of the function Q (P). After the substitution, the student has to solve the equation to find the value of the r. The student has known the values of r in case they have been reading the theory of equation and they have been also knowing that the equation has no solution. After finding the solutions for the r, the students have to use another substitution Q (P) = S(t) and P = et and use the appropriate theorem to change the given differential equation to a constant coefficient equation. After that, the student can use the back substitution to get n independent solutions. The general solution of the given problems can be the set of a linear combination of all linearly independent solutions that can be found by the learners after substituting the above-mentioned equation.

Conclusion

The learners have to change Cauchy’s Euler equation into a constant-coefficient differential formula. Hence, these constant-coefficient equations consist of closed-form solutions that can be solved through Cauchy’s second-degree equation.