First Order Nonlinear D.E.                             

 

 

Example:      Solve the following d.e.          y’  =  dy/dx   =   x2y3  

 

 

 

Strategy:  Note that the Form of D.E. is    dy/dx  = f(x,y)  =  g(x) h(y)

 

So the dependent variable, y, and the independent variable, x, are separable.

 

 

 

   Here   f(x,y)  =  g(x) h(y)  =   x2y3       ( nonlinear because of the term   y3  )

 

 

    So separate the dependent and independent variables using algebra.

 

 

                      dy/ y3    =    x2  dx              Here the variables  y   and   x  are separated.

 

 

 

 

 

   Next integrate once to obtain the solution.  Note:    C1  is the constant of integration.

 

                 ˗  ( ½) y ˗ 2   =  (1/3) x3   +  C1                                               

 

                   1 / y2  =  ˗  (2/3) x3  +  C     

 

                         y  =  √ [  ( 1/ ( C  ˗ (2/3) x3 )  ]                                                (result)

 

 

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