Multi scale expansion of Duffing Equation part IV

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Video series introducing the basic ideas behind perturbation theory. We will cover regular and singular perturbation theory using simple algebraic and ordinary differential equations.The methods are very useful to solve complicated problems, when no exact solutions exist. Some excellent reference books are: • (1) Advanced mathematical methods for Scientists and Engineers, by C. Bender and S. Orszag • (2) Mathematics applied to deterministic problems in natural sciences, by C. Lin and L. Segel. • Acknowledgement: Prof. K.G. Lamb, University of Waterloo for introducing me to the above books. • • In addition, we will cover the basics of complex numbers, differential equations and discuss some applications in Science and Engineering.","styleRuns":[{"startIndex":0,"length":734,"styleRunExtensions":{"styleRunColorMapExtension":{"colorMap":[{"key":"USER_INTERFACE_THEME_DARK","value":4294967295},{"key":"USER_INTERFACE_THEME_LIGHT","value":4279440147}]}},"fontFamilyName":"Roboto"}]},"headerRuns":[{"startIndex":0,"length":734,"headerMapping":"ATTRIBUTED_STRING_HEADER_MAPPING_UNSPECIFIED"}]}},{"itemSectionRenderer":{"contents":[{"continuationItemRenderer":{"trigger":"CONTINUATION_TRIGGER_ON_ITEM_SHOWN","continuationEndpoint":{"clickTrackingParams":"CLcBELsvGAIiEwj7-e3Vg-mLAxVRdHoFHd0hIWA=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/next"}},"continuationCommand":{"token":"Eg0SC3RNejdjRlBaRDVVGAYyJSIRIgt0TXo3Y0ZQWkQ1VTAAeAJCEGNvbW1lbnRzLXNlY3Rpb24%3D","request":"CONTINUATION_REQUEST_TYPE_WATCH_NEXT"}}}}],"trackingParams":"CLcBELsvGAIiEwj7-e3Vg-mLAxVRdHoFHd0hIWA=","sectionIdentifier":"comment-item-section","targetId":"comments-section"}}],"trackingParams":"CLYBELovIhMI-_nt1YPpiwMVUXR6BR3dISFg"}},"secondaryResults":{"secondaryResults":{"results":[{"compactVideoRenderer":{"videoId":"ly4S0oi3Yz8","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/ly4S0oi3Yz8/hqdefault.jpg?sqp=-oaymwEbCKgBEF5IVfKriqkDDggBFQAAiEIYAXABwAEG 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