![]() ![]() When we solve the differential equation P ( x ) y ’’ + Q ( x ) y ’ + R ( x ) y = G ( x ) P(x)y’’ + Q(x)y’ + R(x)y = G(x) P ( x ) y ’’ + Q ( x ) y ’ + R ( x ) y = G ( x ), we’re solving for a function or set of functions y y y that satisfy the following condition: the product of some function P ( x ) P(x) P ( x ) and the second derivative y ’’ y’’ y ’’, plus the product of some function Q ( x ) Q(x) Q ( x ) and the first derivative y ’ y’ y ’, plus the product of some function R ( x ) R(x) R ( x ) and the function y y y, is equal to another function G ( x ) G(x) G ( x ). It can be helpful to verbalize exactly what we’re solving for. Altogether, these terms are equal to G ( x ) G(x) G ( x ), another function of x x x. Each term on the left-hand side of the equation has one of these functions of x x x as its coefficient. We can call this equation linear because each of these functions is a function of x x x only. P ( x ) P(x) P ( x ), Q ( x ) Q(x) Q ( x ), and R ( x ) R(x) R ( x ) are each functions of x x x. Y ’ y’ y ’ indicates the first derivative of y y y with respect to x x x ![]() ![]() ![]() Y ’’ y’’ y ’’ indicates the second derivative of y y y with respect to x x x For example cSolve(m -3m2 +3m2 -3m +2 0,m) will give you the solutions m-ior m-ior m 2or m 1 Solve the following equations making use of these features.P ( x ) y ’’ + Q ( x ) y ’ + R ( x ) y = G ( x ) P(x)y’’ + Q(x)y’ + R(x)y = G(x) P ( x ) y ’’ + Q ( x ) y ’ + R ( x ) y = G ( x ) Use your calculator to find all the roots to the equation both real and complex using cSolve or factor the equation using cFactor. For example, to solve y"-y, + 6y-0 on your calculator use desolve(y"-v, +6y=0,x,y) then enter To solve y"-v, + 6y = 0 with initial conditions y(0)-2 and y"(0)-S On your calculator use desolve(y"-y'6y 0 and y(0) 2 and y(0) 5,x, y) To solve y"-21', + y = on your calculator use desolve(y"-2y, + y = ex / (1 + X2), X, y) To solve higher order homogenous linear differential equations with constant coefficients first create the related polynomial equation. You can also make use of the cFactor and eSolve features to help you solve higher order homogenous linear differential equations. Transcribed image text: Many first and second order differential equations, including initial value problems can be solved using the desolve feature found on your TI-89 calculators. ![]()
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