Calculus- Early Transcendentals, 2021a

10.4. Approximation 385
Exercises for 10.3
In the following exercises, find the general solution of the equation.
Exercise 10.3.1 y ′ + 4y = 8
Exercise 10.3.2 y ′ − 2y = 6
Exercise 10.3.3 y ′ +ty = 5t
Exercise 10.3.4 y ′ + e t y = −2e t
Exercise 10.3.5 y ′ − y = t 2
Exercise 10.3.6 2y ′ + y = t
Exercise 10.3.7 ty ′ − 2y = 1/t, t > 0
Exercise 10.3.8 ty ′ + y = √ t, t > 0
Exercise 10.3.9 y ′ cost + ysint = 1, −π/2 < t < π/2
Exercise 10.3.10 y ′ + ysect = tant, −π/2 < t < π/2
10.4 Approximation
We have seen how to solve a restricted collection of differential equations, or more accurately, how to
attempt to solve them—we still may not be able to find the required anti-derivatives. Not surprisingly,
non-linear equations can be even more difficult to solve. Yet much is known about solutions to some more
general equations.
Suppose φ(t,y) is a function of two variables. A more general class of first order differential equations
has the form y ′ = φ(t,y). This is not necessarily a linear first order equation, since φ may depend on y
in some complicated way; note however that y ′ appears in a very simple form. Under suitable conditions
on the function φ, it can be shown that every such differential equation has a solution, and moreover
that for each initial condition the associated initial value problem has exactly one solution. In practical
applications this is obviously a very desirable property.
Example 10.15: First Order Non-linear
The equation y ′ = t − y 2 is a first order non-linear equation, because y appears to the second power.
We will not be able to solve this equation.