Question

Consider the following statements. (i) Given a second-order linear ODE, the method of variation of parameters gives a particular solution in terms of an integral provided y1 and y2 can be found. (ii) The Laplace Transform is an integral transform that turns the problem of solving constant coefficient ODEs into an algebraic problem. This transform is particularly useful when it comes to studying problems arising in applications where the forcing function in the ODE is piece-wise continuous but not necessarily continuous, or when it comes to studying some Volterra equations and integro-differential equations. (iii) If P and Q are continuous on R, then we can apply the idea of the method of variation of parameters to y′ + P(x)y  =  Q(x) and assume y  =  u1y1 where y1 solves y′ + P(x)y  =  0. That way, we would obtain the same result the integrating factor method would give. (iv) The Laplace Transform of 43tet2 cos(et2) is well-defined for some values of s. (v) To compute solutions to a homogeneous second-order Cauchy-Euler equation, there are three different cases to be considered, depending on whether the characteristic equation associated with the ODE has roots that are real and distinct, or real and equal, or complex. If the equation is non-homogeneous, then we cannot use any method we covered to find a particular solution. (vi) The following is an example of a piece-wise continuous function  f (x)  =  {   1 x   x  ≠  0 0 x  =  0 Determine which of the above statements are True (1) or False (2). Consider the following statements. (i) Given a second-order linear ODE, the method of variation of parameters gives a particular solution in terms of an integral provided yi and y2 can be found. (ii) The Laplace Transform is an integral transform that turns the problem of solving constant coefficient ODEs into an algebraic problem. This transform is particularly useful when it comes to studying problems arising in applications where the forcing function in the ODE is piece-wise continuous but not necessarily continuous, or when it comes to studying some Volterra equations and integro-differential equations. (iii) If P and Q are continuous on R, then we can apply the idea of the method of variation of parameters to y' + P(x)y = Q(x) and assume y = ujyi where yı solves y' + P(x)y = 0. That way, we would obtain the same result the integrating factor method would give. (iv) The Laplace Transform of 43 te cos(e) is well-defined for some values of s. (v) To compute solutions to a homogeneous second-order Cauchy-Euler equation, there are three different cases to be considered, depending on whether the characteristic equation associated with the ODE has roots that are real and distinct, or real and equal, or complex. If the equation is non-homogeneous, then we cannot use any method we covered to find a particular solution. (vi) The following is an example of a piece-wise continuous function XE 0 f(x) X 0 x = 0 Determine which of the above statements are True (1) or False (2).

Answer 1

dy + pdx +242 f(x) Given a second-orden lineap ODE dy dxr Based on the kind of roots of the characteristic equation of lineas ODE, we get different Yit. eg. if ya A enn+Be Berx. Then Y, z Aerix and Y ₂ 2 Be 2x y, and y, are the fundamental solutions of the equation and they are linean independent. Wrons kian, WL9192) YI YL Y Y Then particular solution, 2 W(YINY Yp( x)2 ->, AS WC @ dx + x26] 8 %(%) *(*) dx So Troue (1) Que

SO Frie # # The Laplace Transform is an integral manshorn (by definition). F(s)- Si fcele stat It turns the problems of selvins constant Coefficient des into an algebraic problem Eg y'+2y20 4(0) 23 ^ [y'+2y] 22 [0] =) G[y'+ 2y] 20 2) L [y] + 2 L[y] 0 => [SL [y] - 960)] + 2h [x] 20 =) G[7] 2 3/5+2 a) y (t) 2 3 e 2t (applying invense Laplace tx) # The Laplace tx is applicable for piece-wise continuous funtion, but not neces recessarily continuous hanlion. Voltera canation is x (t) = f(t) + Sk[t,s)x13ds Alinear Veltenna Equation is a convolution equation if xe (t) = f (t) + f k (6-5) x (52ds The frumlion k in the integral is the kennel. such equations can be analyzed and solved by Laplace mansform techniques Similarly, Laplace transform is useful in so true (1) integro - differential aquation. to ges

TS gener linear ODES 111) Variation of parameters methed to solve inhomogeneous fon ist onder inhomogenous linear ODE, it is usually possible to find solutions via integrating factors. y't p(x) y o = Q(x) Integrating facton, I. Fe e SP(x)dx 1.8y' + I PWY = Q(x). I > (@Y.I)2 0(X):) so(x). eSP(x) dx dy Spoex Y selves y'+p(9Y=0 (homogenous eq") dx SO CF 2 7 and general solution is Y CF + PI have $T TO Since it is inhomogenous differenelia so y z uyi So false (2)

leto) valeus of s. because iv) The Laplace Transform of 43 te th cos well-defined for eveny 43 te th cos (et?) is continuous function in IR. and Laplace mansform of every continuous funulion exist so false (2).

x y . So this is dy 2 Y = Dy and dx² ODE.) we have (V) Cauchy a Eulen Equation of ondern has the form anxn M(x) + ani - (x) +.. 007(x)=0 homogenous differential equation if we take dx and dry ID CD-DY So afters turning it to linean homogenous to considers only the Charsacteristic equation. The cases for roots of characterisbic Equation : (1) meal and distinct (1) real and equal (iii) complex for inhomogenous case, we have to find particular solution CPI) by different methods, So True (1)

VO Apiece - wise continuous function is continuous every where except some finite no of points. Home f(x) = { '/* X+0 x=0 so f(x) is continuous every where x20 ER but not at n=o. So f(x2 is piece-wise continuous hunelion So True (1)