TARUN GEHLOT B.E (CIVIL, HONOURS)Qualitative Techniques: Slope FieldsA differentiable function--and the solutions to differential equations better bedifferentiable--has tangent lines at every point. Lets draw small pieces of someof these tangent lines of the function :Slope fields (also called vector fields or direction fields) are a tool tographically obtain the solutions to a first order differential equation. Consider thefollowing example:The slope, y(x), of the solutions y(x), is determined once we know the valuesfor x and y , e.g., if x=1 and y=-1, then the slope of the solution y(x) passingthrough the point (1,-1) will be . If we graph y(x) in the x-y plane, it will have slope 2, given x=1 and y=-1. We indicate this graphically byinserting a small line segment at the point (1,-1) of slope 2.
TARUN GEHLOT B.E (CIVIL, HONOURS)Thus, the solution of the differential equation with the initial condition y(1)=-1 willlook similar to this line segment as long as we stay close to x=-1.Of course, doing this at just one point does not give much information about thesolutions. We want to do this simultaneously at many points in the x-y plane.We can get an idea as to the form of the differential equations solutions by "connecting the dots." So far, we have graphed little pieces of the tangent lines ofour solutions. The " true" solutions should not differ very much from thosetangent line pieces!
TARUN GEHLOT B.E (CIVIL, HONOURS)Lets consider the following differential equation:Here, the right-hand side of the differential equation depends only on thedependent variable y, not on the independent variable x. Such a differentialequation is called autonomous. Autonomous differential equations are alwaysseparable.
TARUN GEHLOT B.E (CIVIL, HONOURS)Autonomous differential equations have a very special property; their slope fieldsare horizontal-shift-invariant, i.e. along a horizontal line the slope does notvary.