Question

toon into EXEL UDL. (d) Choose one of these exact equations (ie., either the ODE you found in part (b) or the one from part (e)) and solve it. Problem 2. Just as we can sometimes turn a non-exact differential equation into an exact one (as in Problem 1), we can sometimes convert a nonlinear ODE to a linear one by making a clever substitution. For example, consider the nonlinear equation (a) Verify that the substitution (t) = (t)) turns Equation (1) into the linear ODE U-20 = -21. (b) Solve the differential equation from part (a) for v(t). (c) Use your answer from part (b) to find the general solution to the original differential equation in Equation (1).

Answer 1

i dove = two-se ..... ci) Take the substitution. 2 = 2 = 82 - Yo. - Putting dari in (1) e n + x = txo = - de +1 = txa = - de + 1 = 219 da 02 - 12 = -27. I .... (ii) out of-edit - ezt Cos Integrating factor is el é zt. is required solution of Ci is - v. e2+ = 1-2t. e-21 dit +C. =-272 Sertat - 18w Jeetat) art]+ 3-21-22+ - - er det dr]+c. = teolt + er2t + C. 2

. 2(t) = t + y + cetzt, But, 21+) - - (22 (+)). r = x + 1 + ceat. (X17))? x(7) = It is required Solution Jetl + ce²t of ().