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Chapter 1
First Order Equations
1.1 Four Examples : Linear versus Nonlinear
A first order differential equation connects a function y.t/ to its derivative dy=dt. That rate of change in y is decided by y itself (and possibly also by the time t).
Here are four examples. Example 1 is the most important differential equation of all.
dy dt
D y 2/
dy dt
D y 3/
dy dt
D 2 ty 4/
dy dt
D y^2
Those examples illustrate three linear differential equations ( 1 , 2 , and 3 ) and a nonlinear differential equation. The unknown function y.t/ is squared in Example 4. The derivative y or y or 2ty is proportional to the function y in Examples 1 , 2 , 3. The graph of dy=dt versus y becomes a parabola in Example 4 , because of y^2. It is true that t multiplies y in Example 3. That equation is still linear in y and dy=dt. It has a variable coefficient 2t, changing with time. Examples 1 and 2 have constant coefficient (the coefficients of y are 1 and 1 ).
Solutions to the Four Examples
We can write down a solution to each example. This will be one solution but it is not the complete solution, because each equation has a family of solutions. Eventually there will be a constant C in the complete solution. This number C is decided by the starting value of y at t D 0 , exactly as in ordinary integration. The integral of f .t/ solves the simplest differential equation of all, with y.0/ D C :
dy dt
D f .t/ The complete solution is y.t/ D
Z (^) t
0
f .s/ ds C C :
2 Chapter 1. First Order Equations
For now we just write one solution to Examples 1 4. They all start at y.0/ D 1.
dy dt
D y is solved by y.t/ D et
dy dt
D y is solved by y.t/ D e t
dy dt
D 2ty is solved by y.t/ D et
2
dy dt
D y^2 is solved by y.t/ D
1 t
Notice : The three linear equations are solved by exponential functions ( powers of e). The nonlinear equation 4 is solved by a different type of function; here it is 1=.1 t/. Its derivative is dy=dt D 1=.1 t/^2 , which agrees with y^2.
Our special interest now is in linear equations with constant coefficients , like 1 and 2. In fact dy=dt D y is the most important property of the great function y D et^. Calculus had to create et^ , because a function from algebra (like y D tn) cannot equal its derivative (the derivative of tn^ is ntn ^1 ). But a combination of all the powers tn^ can do it. That good combination is et^ in Section 1.3.
The final example extends 1 and 2 , to allow any constant coefficient a :
6/ dy dt
D ay is solved by y D eat^ .and also y D Ceat^ /:
If the constant growth rate a is positive, the solution increases. If a is negative, as in dy=dt D y with a D 1 , the slope is negative and the solution e t^ decays toward zero. Figure 1.1 shows three exponentials, with dy=dt equal to y and 2y and y.
e t^ e2t^ et
e
0 t D 1 t
Figure 1.1: Growth, faster growth, and decay. The solutions are et^ and e2t^ and e t^.
