Match with the following A Exact Equation B Integrating Fact

Match with the following
A. Exact Equation
B. Integrating Factor
C. Bernoulli Equation
D. Transient Term
E. Initial Value Problem
F. Linear Equation
G. Autonomous Equation
H. Seperable Equation
I. Homogeneous Substitution
J. Newtons Law of Cooling

Solution

Ans-

e it = cost + i · sin t, t R. 2. ay + by + cy = 0 (a 6= 0) its characteristic equation: ar2 + br + c = 0. 3. Method of Undetermined Coefficients: If in the equation ay + by + cy = g(t), a 6= 0 ´es t I the right-hand side function g(t) has the form g(t) = eut (An(t) cos(vt) + Bm(t) sin(vt)), where An(t), Bm(t) are polynomials of degree n and m respectively, then the particular solution of the inhomogeneous equation has the form: yi,p = t s e ut (Pk(t) cos(vt) + Qk(t) sin(vt)), where s is the multiplicity of the root u + i · v among the roots of the characteristic equation; further, Pk(t) and Qk(t) are polynomials of degree k = max(n, m). 4. Variation of Parameters Method: Consider the inhomogeneous d.e. y + p(t)y + q(t)y = g(t) t I and its homogeneous part Y + p(t)Y + q(t)Y = 0. If the y1, y2 pair is a fundamental solution of the homogeneous d.e., then a particular solution of the inhomogeneous equation is looked for in the form yi,p = C1(t) · y1(t) + C2(t) · y2(t), where for the derivatives of the unknown functions C1(t), C2(t) the following system of equations holds: C 1 (t)y1(t) + C 2 (t)y2(t) = 0 C 1 (t)y 1 (t) + C 2 (t)y 2 (t) = g(t) 5. Special second order d.e.’s: If y is missing, then substitute p(x) := y (x). If x is missing, then substitute q(y) := y 6. The first order d.e. M(x, y)dx + N(x, y)dy = 0 is exact, if M y = N x . To solve the d.e., a function F : R 2 R has to be found such that gradF = (M, N). Then the solution of the d.e. is: F(x, y) = Const.