Example 1 (General Solution):

Determine the general solution to the differential equation dy dx = ( y 1 ) 3 cos ( πx ) .

Identifier: The directions of the problem include the language “ general solution ” and “ differential equation ”.

Step 1: Identify your two variables.

In this example the two variables are the y-variable and x-variable .

d y d x = ( y 1 ) 3 cos ( π x )

Step 2: Separate your equation by variables on to either side of the equals.

1) Multiply both sides of the equation by 1 ( y 1 ) 3 to get the y-variables on the same side as d y .

2) Multiply both sides of the equation by dx to get the x-variables on the same side as d x .

d y d x = ( y 1 ) 3 cos ( π x )

1)                1 ( y 1 ) 3 d y d x = ( y 1 ) 3 cos ( π x ) 1 ( y 1 ) 3

2)                dx 1 ( y 1 ) 3 d y d x = cos ( π x ) dx

 

1 ( y 1 ) 3 d y = cos ( π x ) d x

Step 3: Take the antiderivative of both sides of the equation.

Both antiderivatives require you to run a u-substitution process.

Keep in mind that you only need a single + C that works for the entire equation.

1 ( y 1 ) 3 d y = cos ( π x ) d x

1 ( y 1 ) 3 d y

cos ( π x ) d x

1 ( y 1 ) 3 dy

u = y 1

d u d y = 1

d u = d y

1 ( u ) 3 d u

( u ) 3 d u

u 3 d u = u 2 2

u 2 2 = ( y 1 ) 2 2

cos ( πx ) dx

u = πx

d u d x = π

d u = π d x

d u π = d x

cos ( u ) d u π

1 π cos ( u ) d u

1 π cos ( u ) d u = 1 π sin ( u ) + C

1 π sin ( u ) + C = 1 π sin ( πx ) + C

1 ( y 1 ) 3 d y = cos ( π x ) d x

( y 1 ) 2 2 = 1 π sin ( πx ) + C

Step 4: Solve the equation you found in Step 3 to get the dependent variable ( y -variable) alone on one side of the equals.

( y 1 ) 2 2 = 1 π sin ( πx ) + C

2 ( y 1 ) 2 2 = ( 1 π sin ( πx ) + C ) 2

( y 1 ) 2 = 2 π sin ( πx ) + C

( ( y 1 ) 2 ) 1 2 = ( 2 π sin ( πx ) + C ) 1 2

y 1 = ( 2 π sin ( πx ) + C ) 1 2

y 1 + 1 = ( 2 π sin ( πx ) + C ) 1 2 + 1

y = ( 2 π sin ( πx ) + C ) 1 2 + 1

Final Result:

The general solution to the differential equation dy dx = ( y 1 ) 3 cos ( πx ) would be y = ( 2 π sin ( πx ) + C ) 1 2 + 1 .

 

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