An example where we use initial conditions to solve a repeated-roots differential equation. Watch the next lesson: https://www.khanacademy.org/math/differential-equations/second-order-differential-equations/undetermined-coefficients/v/undetermined-coefficients-1?utm_source=YT&utm_medium=Desc&utm_campaign=DifferentialEquations Missed the previous lesson? https://www.khanacademy.org/math/differential-equations/second-order-differential-equations/complex-roots-characteristic-equation/v/repeated-roots-of-the-characteristic-equation?utm_source=YT&utm_medium=Desc&utm_campaign=DifferentialEquations Differential Equations on Khan Academy: Differential equations, separable equations, exact equations, integrating factors, homogeneous equations. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. ... https://www.youtube.com/watch?v=YezsgIlcb68

At the end of the Mythic Period of China's history, the Xia dynasty gave way to the Shang Dynasty… if the Xia really existed at all! Sal gives an overview of the Xia, Shang, and Zhou Dynasties—societies that lived between the Yellow and Yangtze Rivers—and discusses their cultural practices, their bronzeworking, and the origins of a Chinese identity. Practice this yourself on Khan Academy right now: https://www.khanacademy.org/world-history/e/ancient-china World History on Khan Academy: From prehistory to today, this course covers the human events that have shaped our planet. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything ... https://www.youtube.com/watch?v=g9SGpnheXek Note: All Khan Academy content is available for free at (www.khanacademy.org)

Description ... https://www.youtube.com/watch?v=4C5CdwsUnLw Note: All Khan Academy content is available for free at (www.khanacademy.org)

Transpose of a column vector. Matrix-matrix products using vectors Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transpose/v/linear-algebra-rowspace-and-left-nullspace?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra Missed the previous lesson? https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transpose/v/linear-algebra-transposes-of-sums-and-inverses?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. ... https://www.youtube.com/watch?v=lSoaMCNKfAg Note: All Khan Academy content is available for free at (www.khanacademy.org)