Exercise4.pdf

First- and higher-order ODE Exercises

1. Find the general solution for y(x) using separation of variable, y 0 = X(x)Y (y) → R dy R = X(x)dx. Y (y) (a) y 0 = 3x2 e−y (b) y 0 = 6x2 + 5 (c) y 0 + 4y = 0 (d) y 0 − (y 2 − y)ex = 0 (e) y 0 − 3(y + 1) sin x = 0

2. Solve the initial value problem for x(t) and find limt→∞ x(t). (a) x0 = −2x2 ; x(1) = −1 1 (b) x0 + e−t = 0; x(0) = 3 2 x 0 (c) x = ; x(4) = 8 2t 3. Test if the equation is exact (that is, test if ∂M (x, y)/∂y = ∂N (x, y)/∂x in M (x, y) + N (x, y)y 0 = 0). If exact, solve the initial value problem. (a) x − 2y − (2x − y)y 0 = 0; y(−3) = 5 (b) 3 − y 0 = 0; y(0) = 6 (c) ye−x + 1 + (xe−x )y 0 = 0; y(0) = e (d) 3x2 sin 2y − 2xy + (2x3 cos 2y − x2 )y 0 = 0; y(1) = π/4 (e) 2xy + (y 2 − x2 )y 0 = 0; y(3) = 0 (f) 3x2 y ln y + (x3 ln y + x3 − 2y)y 0 = 0; y(1) = e (g) 4 cos 2x − e−5y y 0 = 0; y(0) = −6 4. Test if the functions in a set are linearly dependent or independent by calculating the Wronskian. (a) {1, x, x2 } (b) {ex , cos x, sin x} (c) {ex , xex , e4x } (d) {ex , e−x , sinh x}

HINT: (sinh x)0 = cosh x, (cosh x)0 = sinh x

5. Find a general solution of the following homogeneous differential equation with constant coefficients. (a) y 00 − 3y 0 + 2y = 0 (b) y 00 − 2y 0 + 3y = 0 (c) y 00 + 2y 0 + 2y = 0 (d) y 00 + 6y 0 + 9y = 0 (e) y 000 + 3y 0 − 4y = 0 (f) y 000 + 5y 00 = 0 (g) y 000 + 3y 00 + 3y 0 + y = 0

6. Find a general solution of the following non-homogeneous differential equation with constant coefficients. (a) y 00 − y 0 = 5 sin 2x (b) y 00 − y 0 = 2xex (c) y 00 − y 0 = x2 ex (d) y 00 + y 0 = 5 − e−x (e) y 00 − 2y 0 + 3y = 2xex + 3 (f) y 00 + y 0 − 2y = 4x2 − 10 sin x (g) y 00 + 2y 0 = 12x2 + 8e2x

Answers 1. (a) y = ln(x3 + C) (b) y = 2x3 + 5x + C (c) y = Ce−4x 1 1 (d) y = ex +C when 0 < y ≤ 1, ex +C when y < 0 or y ≥ 1 e +1 −e +1 (e) y = Ce−3 cos x − 1 2. (a) x(t) = 1/(2t − 3), lim x(t) = 0 t→∞

−t

(b) x(t) = (5 + e )/2, lim x(t) = 2.5 t→∞ √ (c) x(t) = 4 t, lim x(t) = ∞ t→∞

3. (a) x2 + y 2 − 4xy = 94 (b) y = 3x + 6 (c) Not exact (d) x3 sin 2y − x2 y = 1 −

π 4

(e) Not exact (f) x3 y ln y − y 2 = e − e2 (g) 10 sin 2x + e−5y = e30 4. (a) W (b) W (c) W (d) W

= 2, linearly independent = 2ex , linearly independent = 9e6x , linearly independent = 0, linearly dependent

5. (a) y = C1 ex + C2 e2x √ √  (b) y = ex C1 sin ( 2x) + C2 cos ( 2x) (c) y = e−x {C1 sin x + C2 cos x} (d) y = C1 e−3x + C2 xe−3x n  √15   √15 o (e) y = C1 ex + e−x/2 C2 cos x + C3 sin x 2 2 (f) y = C1 e−5x + C2 + C3 x (g) y = C1 e−x + C2 xe−x + C3 x2 e−x 6.

1 cos 2x − sin 2x 2 (b) y = C1 ex + C2 + ex x2 − 2ex x 1 (c) y = C1 ex + C2 + ex x3 − ex x2 + 2ex x 3 −x (d) y = C1 + C2 e + e−x x + 5x √ √  (e) y = ex C1 cos ( 2x) + C2 sin ( 2x) + ex x + 1 (f) y = C1 ex + C2 e−2x − 2x2 − 2x − 3 + 3 sin x + cos x (g) y = C1 + C2 e−2x + e2x + 2x3 − 3x2 + 3x (a) y = C1 ex + C2 +