Question

$4.1, Exercise 5. Solve the differential equation dt = 2x, where x(0) = 1. At what time ty will altı) = 2?

Answer 1

function of af Recall Iht. The genere solution of Linear differentia Equation du pp(a). Y = Qua) i y. (IF) = |ax (IF). Ante where Ifa Integrating facty=en Spen) an. yra on where a constant.) Her y has {a independent versable y ů dependent varianh sol. Given differential equation in 21 = 22. dre - 2x=0 Here Plt)=-2 : Q(t)=0 Now Integrations factu: * = { petat 2 at 2t. Now. The general solution of differentil Equah. x. ezt seco) at a c) x-dependent. -xachat E xis cet 21a - Now Initial condition given, Shent x= 1 whanto Hence san o yield. = c.ro) for) c=1. . Their fore doon from Eqnco). the required solution i x = 27 ) @ Now geven that alt,) = 2 : when t=t. from Eqh (2) = 2= ezti oth = 2 aty=1092 ti= loge when taloge " selt)=2

Given 100g differential equation i ane azn seperating the variable del = 2.21 Inlegrating the bolt sider loge zette x = 2t+C et e .e x = x = 2 C AL e - consland) now the condition. apply intich XCO)=1 no weget c=1 } = e.cz ? Can J The required soluton & O . 2 e 7 Now find the time: t, when alt=2 From : 2 = 2t, a 10g2a fat, t = log2 The time ty = loge, where alt;)=2