TECHNIQUES FOR SOLVING DIFFERENTIAL EQUATIONS Separable Equations dy dx =g (x)f(y) Step 1: Separatex’s andy ’s on different sides. 1 f(y) dy= g (x)dx Step 2: Integ rate both sides. 1 f(y) dy = g(x)dx + CSt ep 3: Expres s y in terms ofxwhere possible. If | y| =h(x), then y = ±h(x). Ify = ±e Ch(x), then y = A h(x) where Ais anyreal number (including zero). Ste p 4: Che ck that constant solu tionsy = Cwheref(C) = 0 are not missed. First Order Linear Equations y + P(x)y= Q(x) Step 1: Find i nt egr ati ng f act or. I(x) =e RP(x)dx Step 2: Write differ ential equat ion as I(x)y =I(x)Q(x) Step 3: Integ rate both sides. I(x)y= I(x)Q(x)dx + CSt ep 4: Di vi de bot h si des byI(x) y = 1 I(x) I(x)Q(x)dx + CRemark: Be careful with the sign ofP(x). For instance, Ify + 1 x y= 1, the integrating factor is I(x) =e R1/x dx =x. Ify − 1 x y= 1, the integrating factor is I(x) =e R −1/x dx = 1 x . Second Order Linear Homogeneous Equations ay + by + cy= 0 Step 1: Write down the aux iliary equat ion. ar 2 + br+ c= 0 Step 2: Sol ve the auxili ary equation. r = −b ± √ b 2 − 4ac 2a Step 3: (i) (ii) (iii) Depending on the roots: b 2 − 4ac >0. Two real roots r 1 ,r 2 . b 2 − 4ac= 0. One real root r = r 1 = r 2 . b 2 − 4ac <0. Two complex roots α ± iβ . y = c 1 e r1x + c 2 e r2x y = c 1 e rx + c 2 x e rx y = c 1 e αx cos βx+ c 2 e αx sin βx
