Cheat Sheet – Differential Equations Variable Separable Reducible to Variable Separable Form Form dy = F(x). G(y) dx dy = f(ax + by + c) dx Solution Solution dy Separate the variables: = F(x)dx G(y) Substitute ax + by + c = t ⇒ a + b Integrate both sides: ∫ dy dt = dx dx 1 dt ( − a) = f(t) b dx dt Separate the variables: = dx bf(t) + a dy = ∫ F(x)dx G(y) Convert the equation: Integrate both sides: ∫ dt = ∫ dx bf(t) + a Homogeneous Reducible to Homogeneous Form Form dy y = f( ) dx x dy ax + by + c = ; dx px + qy + r Solution Solution dv dy Put y = vx ⇒ v + x = dx dx Put x = X + h, y = Y + k dv = f(v) dx dv dx Separate the variables: = f(v) − v x where ah + bk + c = 0, ph + qk + r = 0 Convert the equation: v + x Integrate both sides: ∫ Convert the equation: dv dx =∫ f(v) − v x a b ≠ p q dY aX + bY = dX pX + qY Solve as a homogeneous differential equation Linear Reducible to Linear Form Form dy + P(x)y = Q(x) dx f′(y) dy + P(x)f(y) = Q(x) dx Solution Solution Calculate Integrating Factor: I(x) = e∫ P(x)dx Substitute f(y) = t ⇒ f ′ (y) Solution: y. I(x) = ∫ Q(x). I(x)dx Convert the equation: dy dt = dx dx dt + P(x)t = Q(x) dx Solve as a linear differential equation © www.doubleroot.in