Introductory Differential Equations using Sage - William Stein

1.4. FIRST ORDER ODES - SEPARABLE AND LINEAR CASES 31
∫
µx =
Dividing both side by µ gives (1.6).
∫
(µx) ′ dt =
µq(t)dt.
Exercises:
Find the general solution to the following seperable differential equations:
1. x ′ = 2xt.
2. x ′ − xsin(t) = 0
3. (1 + t)x ′ = 2x.
4. x ′ − xt − x = 1 + t.
5. Find the solution y(x) to the following initial value problem: y ′ = 2ye x , y(0) = 2e 2 .
6. Find the solution y(x) to the following initial value problem: y ′ = x 3 (y 2 +1), y(0) = 1.
7. Use the substitution v = y/x to solve the IVP: y ′ = 2xy
x 2 −y 2 , y(0) = 2.
Solve the following linear equations. Find the general solution if no initial condition
is given.
8. x ′ + x = 1, x(0) = 0.
9. x ′ + 4x = 2te −4t .
10. tx ′ + 2x = 2t, x(1) = 1 2 .
11. dy/dx + y tan(x) = sin(x).
12. xy ′ = y + 2x, y(1) = 2.
13. y ′ + 4y = 2xe −4x , y(0) = 0.
14. y ′ = cos(x) − y cos(x), y(0) = 1.
15. In carbon-dating organic material it is assumed that the amount of carbon-14 ( 14 C)
decays exponentially ( d 14 C
dt
= −k 14 C) with rate constant of k ≈ 0.0001216 where
t is measured in years. Suppose an archeological bone sample contains 1/7 as much
carbon-14 as is in a present-day sample. How old is the bone?
16. The function e −t2 does not have an anti-derivative in terms of elementary functions,
but this anti-derivative is important in probability. So we define a new function,
erf(t) := √ 2 ∫ u
π 0 e−u2 du. Find the solution of x ′ − 2xt = 1 in terms of erf(t).