Differential Equations Workbook For Dummies®

1.3. Solving Linear First Order Differential Equations That Involve Terms in y

Wondering what to do if a differential equation you're facing involves both x and y?

Start by taking a look at this representative problem:

The preceding is a linear first order differential equation that contains both dy/dx and y. How do you handle it and find a solution? By using some algebra, you can rewrite this equation as

Multiplying both sides by dx gives you

Congrats! You've just separated x on one side of this differential equation and y on the other, making the integration much easier. Speaking of integration, integrating both sides gives you

ln |y − (b/a)| = ax + C

where C is a constant of integration. Raising both sides to the power e gives you this, where c is a constant defined by c = e^C:

y = (b/a) + ce^ax

Anything beyond this level of difficulty must be approached in another way, and you deal with such equations throughout the rest of the book.

If you think you have solving linear first order differential equations in terms of y all figured out, try your hand at these practice questions. ...