How are Linear Differential Equations formed?A linear differential equation of second or higher order have more than 1 solutions of y ( y 1 , y 2 , etc.). Hence a common solution is obtained by assuming y = c 1 y 1 + c 2 y 2 + . . . . . Now my questions are:1. How are linear differential equations formed from general algebraic expressions?2. Why is it that a linear differential equation of second or higher order contain more than one solution of y that satisfy them?

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How are Linear Differential Equations formed?A linear differential equation of second or higher order have more than 1 solutions of   y  (  y  1  ,  y  2  , etc.). Hence a common solution is obtained by assuming   y  =  c  1  y  1  +  c  2  y  2  +  .  .  .  .  . Now my questions are:1. How are linear differential equations formed from general algebraic expressions?2. Why is it that a linear differential equation of second or higher order contain more than one solution of y that satisfy them?

trestegp0 · Accepted Answer

Step 1If you&#039;re good with your linear algebra, solutions to an       n          t      h       order linear differential equation form a vector space of dimension n, and every vector space has a basis of n linearly independent vectors that span the space. In this case our vectors are functions!Step 2So an       n          t      h       order equation will form an n-parameter family of solutions that are closed under addition and scalar multiplication. So if       y    1  ,       y    2  ,...,       y    n   are solutions to the equation then       c    1        y    1    +      c    2        y    2    +...+      c    n        y    n   is also a solution for any real numbers       c    1    ,      c    2  ,...,      c    n  . This is also sometimes called the &quot;principle of superposition&quot;.

Makayla Reilly · Accepted Answer

Step 1When we differentiate f(x) and   f  (  x  )  +  c, for any constantc we get the same derivative. If we are given that derivative we cannot say with definiteness for which functio9ns it is the derivative.That is the function cosx could be the derivative of either sinx or   sin  ⁡  x  +  8.Step 2Now second order differential equation involves the second derivative: So the two functions   −  cos  ⁡  x  +  8  x  ,  −  cos  ⁡  x both when differentiated twice could give sinx. These two are solutions of       y    ″    =  cos  ⁡  x and are linearly independent.To form a a linear differential equation from an algebraic expression is easy; The derivative of   sin  ⁡  (  x  −  a      )    2   is   2  (  x  −  a  )  cos  ⁡  (  x  −  a      )    2  . Now we have to find an algebraic relationship between the function   y  =  sin  ⁡  (  x  −  a      )    2   and its derivative:  2  (  x  −  a  )  (  1  −      y    2    )  −      y    ′    =  0

How are Linear Differential Equations formed? A linear differential equation of second or higher order have more than 1 solutions of y(y_1, y_2, etc.). Hence a common solution is obtained by assuming y=c_1y_1+c_2y_2+.....

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