978-3-319-17852-3

10 1. First-Order Differential Equations
a) x ′ = 2x t , x = t2 . b) x ′ = − t x , x = √ 6 − t 2 .
2. Verify the solutions to the initial value problems in Example 1.2.
3. Which of the following functions,
is a solution to the DE x ′ = −x 2 ?
4. Show that both
x(t) = 1 t , x(t) =2 t , x(t) = 1
t − 2 ,
x 1 (t) =e −t cos t and x 2 (t) =e −t sin t
are solutions to the second-order differential equation
Show that
x ′′ +2x ′ +2x =0.
x(t) =Ae −t sin t + Be −t cos t
is a solution for any values of the constants A and B.
5. Show that x(t) = ln(t + C) is a one-parameter family of solutions, or
integral curves, of x ′ = e −x ,whereC is an arbitrary constant. Plot the
integral curves using the values of C given by C = −2, −1, 0, 1, 2. On the
plot indicate the particular solution that satisfying x(0) = 0.
6. Find a solution x = x(t) of the equation x ′ +2x = t 2 +4t + 7 in the form
of a quadratic function of t, that is, of the form x(t) =at 2 + bt + c, where
a, b, andc are to be determined.
7. Find values of m for which x(t) =t m is a solution to 2tx ′ = x.
8. Find values of m for which x(t) =t m is a solution to t 2 x ′′ − 6x =0.
9. Find two values of λ for which x(t) =e λt is a solution of the differential
equation 2x ′′ − 5x ′ − 3x =0.
10. Show that the one-parameter family of straight lines x(t) =Ct + f(C) is
a solution to the differential equation tx ′ − x + f(x ′ ) = 0 for any value of
the constant C.
11. Plot the one-parameter family of curves x(t) =(t 2 − C)e 3t for different
values of C. Find a differential equation whose solution is x = x(t). Hint:
Find x ′ and then obtain a relation between t, x, andx ′ .