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nt) y = 2t, y(1) = 2 y(t) = 0 (4-1²) y' + 2ty = 3t², y(-3) = 1 2. y' + (tant) y = sint, 3. 4. (Int) y' + y = cott, y(2) = 3 In each of Problems 5 through 8, state where in the ty-plane the hypotheses of Theorem 2.4.2 are satisfied. nensilid auom 5. y'=(1-1² - y2) 1/2 6. y'= 7. 8. y': 1+1² 3y-y² In each of Problems 9 through 12, solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value Yo. In |ty| 1-1² + y² y' = (1² + y²) 3/2 no 9. y'= -4t/y, y(0) = yo 10. y' = 2ty², y(0) = yo (0) de G G E asimsnya noit

nt) y = 2t, y(1) = 2 y(t) = 0 (4-1²) y' + 2ty = 3t², y(-3) = 1 2. y' + (tant) y = sint, 3. 4. (Int) y' + y = cott, y(2) = 3 In each of Problems 5 through 8, state where in the ty-plane the hypotheses of Theorem 2.4.2 are satisfied. nensilid auom 5. y'=(1-1² - y2) 1/2 6. y'= 7. 8. y': 1+1² 3y-y² In each of Problems 9 through 12, solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value Yo. In |ty| 1-1² + y² y' = (1² + y²) 3/2 no 9. y'= -4t/y, y(0) = yo 10. y' = 2ty², y(0) = yo (0) de G G E asimsnya noit

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Differential Equations Problems** In each of Problems 1 through 4, determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. 1. \((t - 3)y' + (\ln t)y = 2t,\quad y(1) = 2\) 2. \(y' + (\tan t)y = \sin t,\quad y(\pi) = 0\) 3. \((4 - t^2)y' + 2ty = 3t^2,\quad y(-3) = 1\) 4. \((\ln t)y' + y = \cot t,\quad y(2) = 3\) --- In each of Problems 5 through 8, state where in the \(ty\)-plane the hypotheses of Theorem 2.4.2 are satisfied. 5. \(y' = (1 - t^2 - y^2)^{1/2}\) 6. \(y' = \frac{\ln |t|}{1 - t^2 + y^2}\) 7. \(y' = (t^2 + y^2)^{3/2}\) 8. \(y' = \frac{1 + t^2}{3y - y^2}\) --- In each of Problems 9 through 12, solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value \(y_0\). 9. \(y' = -4t/y,\quad y(0) = y_0\) 10. \(y' = 2ty^2,\quad y(0) = y_0\) 11. \(y' + y^3 = 0,\quad y(0) = y_0\) 12. \(y' = \frac{t^2}{y(1 + t^3)},\quad y(0) = y_0\)
Transcribed Image Text:**Differential Equations Problems** In each of Problems 1 through 4, determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. 1. \((t - 3)y' + (\ln t)y = 2t,\quad y(1) = 2\) 2. \(y' + (\tan t)y = \sin t,\quad y(\pi) = 0\) 3. \((4 - t^2)y' + 2ty = 3t^2,\quad y(-3) = 1\) 4. \((\ln t)y' + y = \cot t,\quad y(2) = 3\) --- In each of Problems 5 through 8, state where in the \(ty\)-plane the hypotheses of Theorem 2.4.2 are satisfied. 5. \(y' = (1 - t^2 - y^2)^{1/2}\) 6. \(y' = \frac{\ln |t|}{1 - t^2 + y^2}\) 7. \(y' = (t^2 + y^2)^{3/2}\) 8. \(y' = \frac{1 + t^2}{3y - y^2}\) --- In each of Problems 9 through 12, solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value \(y_0\). 9. \(y' = -4t/y,\quad y(0) = y_0\) 10. \(y' = 2ty^2,\quad y(0) = y_0\) 11. \(y' + y^3 = 0,\quad y(0) = y_0\) 12. \(y' = \frac{t^2}{y(1 + t^3)},\quad y(0) = y_0\)
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