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LINEAR ALGEBRA, diffrential equations, initial and boundary value problem,…

- - - - ROW REDUCED ECHELON FORM
        
        ALL THE PIVOT ELEM ARE 1
        
        ONLY NON ZERO ELEM IN A PIVOT COLUMN ARE THE PIVOT ELEM
        
        unique form
      - LEADING TERMS RELATIVE POSITIONS
      - ZERO ROWS AT THE BOTTOM
        
        DOUBT IF TWO ZERO ROWS ???
  - - - matrices obtained by performing a single row operation on a identity matrix having the same number of the rows
      - performing a row operations on matrix is equivalent to multiplying the matrix by the corresponding elementary matrices in the same order
  - - - of upper triangular or lower triangular or diagonal matric
        
        product of the diagonal elements
      - use in cramer rule
      - use for inverse
      - find the adjoint
  - - - same characteristic polynomial
- - - - VECTOR ADDITION
      - SCALAR MULTIPLICATION
      - DISTRIBVUTIVE LAW
    - - ITS COMPLETELY UPTO US HOW WE DEFINE THE OPERATIONS TILL THE PROPERTIES WITH RESPECRTT TO THOSE OPERATIONS ARE STAISFIED
  - - - LINEAR SPAN OF A SET
        
        LINEAR COMBINATION OF VECTORS
        
        LINEARLY INDEPENDENT SET OF VECTORS VS DEPENDENT
        
        BASIS SET
        
        ORDERED BASIS
        
        DIMENSIONS OF A VECTOR SPACE
        
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        ORDERED BASIS
        
        CHANGE OF BASIS
        
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    - - DOUBT DOES IT MEAN THAT UNION OF A SUBSPACE AND A SUBSPACE OF THE SUBSPACE IS A SUBSPACE
        
        ANSWER IS NO
    - - SUM OF TWO SUBSPACES WHICH DO NOT HAVE A INTERSECTION
- - - - SATISFIES
        
        L(AX+BY) =AL(X) + BL(Y)
  - - - FOR HIGHER ORDER LINEAR HOMOGENOUS ODE NOT FOR ANY HIGHER ORDER ODE
        
        continuity AND BOUNDED of the functions whcih are the coefficients in an open interval
        
        unique solution in that open interval
        
        BUT THE THE COEFFICIENT OF THE HIGHEST ORDER TERM BE ONE
      - V
    - - giiven the inital value of the function and its derivative at one point
      - HOW TO SOLVE
        
        solve by change of variable and conversion to a system of 1st order equation
    - - general solution
        
        must have two independent parameters
        
        we can form the general solution from two independent solutions
        
        INDEPEDNT and DEPENDENT SOLUTIONS
        
        THE FUCNTIONS CORR. TO THESE SOLUTIONS ARE INDEPEDNT LINEARLY VECTORS
        
        TO CHECK INDEPENDENCE USE THE WRONSKIAN
        
        BUT THE WRONSKIAN ONLY FOR TWO FUNCTIONS
        
        ZERO WRONSKIAN MEANS L .D NO !!!!!!!!!
        
        NOTE THE ABELS FORMULA FOR THE WROSKIAN ONLY VALID WHEN THE COEFFIECIENT OF THE HIGHEST ORDER DERIVATIVE IS 1
        
        ONLY WHEN THE TWO FUNCTIONS CONSIDERED ARE THE SOLUTION OF A LINEAR ODE THEN WRONSKIAN ZERO IMPLIES THAT L.D
        
        wronskian
        
        determinant
        
        n th row has the (n-1) derivative of the functions
        
        order in the wroskian matter W(x1,x2) <>W(x2,x1)
        
        SPECIAL PROPERTIES IF FUNCTION ARE SOLUTION OF AN ODE
        
        ABELS THEOREM
        
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        IT BEING ZERO IS SUFFICIENT FOR LD
        
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        binary behavior
        
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        FUNDAMENTAL SET OF SOLUTIONS
        
        any solution can be written as a LC of fundamental set of solutions
        
        Akin to the basis set we have in vector spaces
        
        Order of ODE is like the dimension of the vector space and decides the number of functions in the fundamental set
        
        TRICK TO PROVE LD OR LI
        
        IF LD THEN ONE IS THE CONSTANT MULTIPLE OF OTHER
        
        AND USE THAT DERIVATIVE OF CONSTANT ZERO
  - - - homogenous
        
        constant coeffiecnt
        
        put the trial solution for the exponential
        
        find the charactertic equation
        
        find its roots
        
        repeated roots
        
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        complex roots
        
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        note find all the roots real or complex dont forget the complex ones
        
        non constant coeff
        
        check if eulers equation
      - non homo
        
        VPM method
        
        solve the corr homo
        
        undetermined coefficient method
        
        CORE idea
        
        try a solution based on the look part which make equation non homo
        
        but if your guess matches some part of the GE OF THE HOMO then multiply by t
        
        doubt WHY
        
        IMP RELATION WITH THE CORR HOMO
        
        general solution of the non homo is the sum of
        
        particular sol of non homo
        
        general sol of the homo
        
        also implies that the difference of any two solutions of the non homo equation is a solution of the homo
        
        DOUBT
        
        any two solution will work ??
        
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- - - - USE THE PROPETIES TO GET AN ALEGBRAIC EQUATION IN TERMS OF THE LAPLACE TRANFORM OF THE DPENDENT VARIABLE
        
        THE SOLUTION TO THE DE IS NOTHING BUT THE INVERSE LAPLACE TRANSFORM OF THE LAPLACE TRANSFORM OF THE DPEENDNET VARIABLE
        
        USE THE INVERSE LAPLACE TRANSFORM PROPERTIES IF NEEDED
  - - - WHEN PIECEWISE CONTINOUS AND EXPOENTIAL ORDER FUNCTION
    - - REMEBER THE LAPLACE TRANSFORM OF COMMON FUNCTIONS
        
        TRIGO
        
        HYPERBOLIC
        
        EXPONETIAL
        
        DIRAC DELTA
        
        STEP FUNCTION
        
        POLYNOMIAL
      - FUNCTION MULTIPLIED WITH AN EXPONENTIAL
      - LINAERITY AND THE
      - DIFFERENTIATION
        
        OF THE LAPLACE TRANSFORM
        
        OF THE FUNCTION WHOSE LAPLACE TRANSFORM WE ARE FIDNING
      - INTEGRATION
      - FUNCTION MULTIPLIED WITH AN HEAVSIDE STEP FUNCTION
    - - PARTAIL FRACTIONS
  - - - EXPOENTIAL MULTIPLY SOEM FUNCTION
      - CONVOLUTION
      - PARTAIL FRACTION
        
        POLYNOIMAL BY POLYNIMAL
      - INTEGRATION
      - DIFFERENTIATION
      - ANY OTHER PROPERTY OF THE LAPCAE TRANSFORM JUST TAKE INVERSE BOTH SIDES
- - - - order
        
        first order
        
        IVP in a particular form
        
        solutions
        
        unique
        
        all the conditions for the existence but also the lipschitz wrt to dependent variable in the rectanguar region IMP in that region only
        
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        NOW WHAT IF THE FUNCTION ID NOT LIPSHITZ BUT CONTINOUS
        
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        LIPSCHITZ
        
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        existence
        
        THM based on the continuity of the function of the dependent and independent variable in a rectangular region centered about the given initial point
        
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        NOTE THAT IF THE HYPOTHESIS OF THE THM NOT MEET THEN WE CANNOT SAY THAT NO SOLUTION
        
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        Global uniqueness THM
        
        gurantees the unique solution in the interval where continuty is there
        
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        importance to the lipchitzness
        
        meaning of the lipschitz wrt to one variable
        
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        NOTE THE CONDITIONS FOR THE EXISTENCE AND UNIQUENESS THEOREM ARE SUFFICIENT BUT NOT NECESSARY
        
        HOW TO SOLVE AN IVP
        
        PICARDS ITERATION METHOD
        
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        well defined IVP
        
        unique solution
        
        solution depends continuous on the inital value OR stablility REALTED TO THE CS conditioning i think so
        
        EXAMPLES WERE THIS NO STABILITY ????
        
        idea is that the if the error in initall value is less than the error in the solution is also less
        
        solution exists
        
        Solving first order ODEs
        
        separation of variables
        
        exact DE
        
        converting to exact
        
        solve for the independent variable in terms of dependent find x as a function of y
        
        DIFFEREENTAITE ONCE MORE CLAURAUTS EQ
        
        THEN BACK SUBSTITUE
      - degree
    - - NEVER FORGET THE MODULUS WHEN INTEGRATING AND GETTING LOGARITHMIC FUNCTION
      - SOMETIMES WHILE SOLVING AN ODE YOU NEED TO DIFFERENTIATE IT ONCE MORE
        
        REMEMBER CLAURAIUTS EQUATIONS