Test 1 Solutions

# Test 1 Solutions - ty-plane represented by D(c Yes No Does...

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MAT 274 TEST 1 Instructor : Dongrin Kim Name : Score : 1. Find an integrating factor for the linear diﬀerential equation ty 0 + ( t + 1) y = t, t > 0. Ans : te t 2. Find the general solution for y 0 +(2 /t ) y = (cos t ) /t 2 . Use the integrating factor μ ( t ) = t 2 . Ans : y ( t ) = sin t + c t 2 1

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3. Find the solution of y 0 + (2 /t ) y = 3 + 1 t 2 ( t 2 +1) , y (1) = π/ 4 + 2 , t > 0. Use the integrating factor t 2 . Ans : y ( t ) = t + tan - 1 ( t ) + 1 t 2 4. The general solution to a diﬀerential equation y 0 + p ( t ) y = g ( t ) , t > 0 is y ( t ) = 4( t + 1) - 2 e 3( t - 1) - 2cos( t - 1) + c ln( t + 1) . (a) Determine how the solution behave as t → ∞ . Ans : y → ∞ as t → ∞ (b) Find c for the initial value problem y (1) = 0. Ans : c = ln2 2
5. Find the implicit solution for y 0 = 3 x 2 + 4 x + 2 2( y - 1) . Ans : y 2 - 2 y = x 3 + 2 x 2 + 2 x + c 6. Find all constant solutions for y 0 = ( x 2 - 1)ln( y + 2) y 2 + 3 y . Do not solve the problem. Ans : y = - 1 3

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7. Find the interval in which the solution to ( t 2 - 1) y 0 + y = tan t, y (1 . 5) = 3 is certain to exist. Use the interval notation. Do not solve the diﬀerential equation. Ans : (1 ,π/ 2) 8. f ( t,y ) = (1 - t 2 - y ) - 1 / 2 (a) Find the set D = { ( t,y ) | ··· } such that f is continuous. Ans : D = { ( t,y ) | y > t 2 - 1 } (b) Sketch the region in the
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Unformatted text preview: ty-plane represented by D . (c) ( Yes , No ) Does the theorem guarantee the unique solution to y = f ( t,y ) , y (0) = 0 ? Ans : Yes, (0 , 0) is inside the region D . 4 9. Solve ( y/x + 6 x ) dx + (ln x-2) dy = 0. This is an exact diﬀerential equation. Give an implicit solution. Ans : y ln x + 3 x 2-2 y = c 10. Find an integrating factor for e x dx + ( e x cot y + 2 y csc y ) dy = 0. Do not solve the equation. (Hint: μ = M y-N x N μ or μ = N x-M y M μ ) Ans : μ ( y ) = (cot y ) μ ( y ) ⇒ μ ( y ) = sin y 11. y = y (3-ty ) , y (0) = 0 . 5. Use euler’s method to ﬁnd approximation for the solution φ ( t ). Use the time interval h = 0 . 2. (a) φ (0 . 2) ≈ Ans : . 5 + [0 . 5(3-· . 5)] · . 2 = 0 . 53 (b) φ (0 . 4) ≈ Ans : . 53 + [0 . 53(3-. 2 · . 53)] · . 2 = 0 . 836764 5...
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Test 1 Solutions - ty-plane represented by D(c Yes No Does...

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