Differential equations in functionEquations (1): x y ′ + ( 1 − x ) y = 1determine and solve the differential equation (2) whose general solution is the function z .determine the general solution of (1)

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Differential equations in functionEquations (1):   x      y    ′    +  (  1  −  x  )  y  =  1determine and solve the differential equation (2) whose general solution is the function z .determine the general solution of (1)

soporoseun · Accepted Answer

Step 1Compute       z    ′    =  y  +  x      y    ′    =  y  +  1  −  (  1  −  x  )  y  =  zSo the general solution of the above equation (call it eqn (2)) is  z  (  x  )  =  z  (  0  )      e    x    =      e    x  Step 2because   z  (  0  )  =  1. Solving then for y:  y  (  x  )  =            z      −      1        x    =                    e        x            −      1        x

Gillian Cooper · Accepted Answer

Step 1If   z  =  x  y  +  1 then                                          d            z                                d            x                                              =                              d                      d            x                                    (          x          y          +          1          )                                                  =                    y        +        x                              d            y                                d            x                              Step 2Now:                    x                              d            y                                d            x                          +        y        −        x        y                            =                    1                                    ⟹                x                              d            y                                d            x                          +        y                            =                    1        +        x        y                                    ⟹                                      d            z                                d            x                                              =                    z            Now you solve this equation for z and then change back the variables to x and y at the end.

Differential equations in function. Equations (1) : xy′+(1-x)y=1 let z=xy+1

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