TEKNILLINEN TIEDEKUNTA, MATEMATIIKAN JAOS
DIFFERENTIAALIYHTÄLÖT
1. välikoe 1.3.2008
Read problems carefully. Write down your calculations. Check your solution.
1.
Find the general solution to the differential equation x3 y ′ − 2x2 y = y −2 (x > 0). Find
the solution, which satisfies the initial condition y(1) = 1. Use the substitution of the form
z = y 1−k .
2.
a) Solve the integral equation
Rx
y(x) = 1 + (y(t))2 t3 dt,
0
by using the Picard-Lindelöf method. Use the initial guess ϕ0 (x) = 0 and compute three
iterations i.e. compute ϕ1 (x), ϕ2 (x), ϕ3 (x).
b) Find the exact solution for the previous integral equation
y(x) = 1 +
Zx
(y(t))2 t3 dt.
0
Find the corresponding initial value problem for a differential equation and solve it. Use the
Rx
differentiation formula D f (t)dt = f (x).
a
3.
a) Derive the formula for integration by parts by starting from the formula of the derivative
of a product.
(1p)
−sM
b) Let the limit lim f (M )e
= 0 exist. Derive the formula for the Laplace transform for
M →∞
the derivative of a function f by means of integration by parts.
(1p)
c) Solve by using Laplace transform the initial value problem y ′′ +8y ′ +25y = 26e−2t , y(0) = 3,
y ′ (0) = 7.
(4p)
4.
Lecturer M. Hamina has some problems. His students cannot differentiate and integrate properly. For the students the processes of differentiation and integration interchange in a random manner. In addition students do not know calculation rules for elementary functions.
They calculate with almost all real valued functions of one real variable (as for example
the square root, the exponential function, the inverse number, etc ...) by using the rule
f (x + t) = f (x) + f (t), for all x, t ∈ Mf . Help lecturer Hamina to construct for exams such
differential equations, which his students can solve.
a) Find differentiable functions such that the interchange between differentiation and integration makes no harm.
b) Express the definition of the derivative and find differentiable functions f : R → R, such
that the rule f (x + t) = f (x) + f (t), for all x, t ∈ R is valid.
Taulukko 1. Functions and Laplace–transforms
f (t)
1
t
F (s)
tn−1
(n−1)!
f (t)
sin ωt
cos ωt
eat
1
s
1
s2
1
sn
F (s)
ω
s2 +ω 2
s
s2 +ω 2
1
s−a
L(ect f (t)) = F (s − c)
L(tn f (t)) = (−1)nF (n) (s)
L(H(t − c)f (t − c)) = e−cs F (s))
L(δ(t − c)) = e−cs
L(f ) = F (s) =
Z∞
0
f (t)e−stdt