Lecture Notes in Differential Equations - Bruce E. Shapiro

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Definition 1.2 (Solution, ODE). A function y = φ(t) is called a solution
of y ′ = f(t, y) if it satisfies
φ ′ (t) = f(t, φ(t)) (1.9)
By a solution of a differential equation we mean a function y(t) that satisfies
equation 1.3. We use the symbol φ(t) instead of f(t) for the solution
because f is always reserved for the function on the right-hand side of 1.3.
To verify that a function y = f(t) is a solution of the ODE, is a solution,
we substitute the function into both sides of the differential equation.
Example 1.2. A solution of
is
dy
dt
= 3t (1.10)
y = 3 2 t2 (1.11)
We use the expression “a solution” rather than “the solution” because solutions
are not unique! For example,
y = 3 2 t2 + 27 (1.12)
is also a solution of y ′ = 3t. We say that the solution is not unique.
Example 1.3. Show 1 that y = x 4 /16 is a solution of y ′ = xy 1/2
Example 1.4. Show 2 that y = xe x is a solution of y ′′ − 2y ′ + y = 0.
Example 1.5. We can derive a solution of the differential equation
by rewriting it as
dy
dt = y (1.13)
dy
y
= dt (1.14)
and then integrating both sides of the equation:
∫ ∫ dy
y = dt (1.15)
1 Zill example 1.1.1(a)
2 Zill example 1.1.1(b)