Let V be an inner product space and let W be a fixed veclor in V. Let L: V -+ R be defined by L (v) = (, .. w) for V in V. Show that L is a linear transfonnation.

Math246 Lecture 3: Separable Equations This is our first look at nonlinear first order differential equations. Separable equations take the dy form P (y)dx=W (x) . Solving these equations is fairly straightforward. We will take a 3 step approach. 1. Rewrite into the form P(y)dy=W (x)dx 2. Integrate both sides: P y dy= W(x)dx ∫ ( ) ∫ 3. Solve for the general or explicit solution. There’s not much left to say except let’s work through a few examples, so let’s work through a few examples! Examples dy 3 1. Solve =7x y dx Ok, first we move everything into place. 1 3 dy=7x dx y Next, we integrate both sides. 1 3 ∫ dy=∫7x dx y 3 7 x . The integral of 1/y is the natural log function and the integra is 4 . l|y|= 7 x +C 4 y And now we solve for by taking the exponential of both sides. 7 4x y=Ae eC Where A is the constant Let’s work through one with a condition. 2x −x−1 f(x)= 3 y(1)=15 2. 2y −6 , Ok, first we rewrite it into a proper form. 2 x (¿¿4−x−1)dx 3 (2y −6)dy=¿ And now we