Homework 6 - pacheco(jnp926 Homework 6 staron(52840 This...

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pacheco (jnp926) – Homework 6 – staron – (52840) 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001(part1of3)10.0points For the differential equation dy dx + 3 y = 8 e 3 x , (i) first find its general solution. 1. y = 4 3 e 3 x + C e 3 x correct 2. y = 4 3 e 3 x + Ce 3 x 3. y = 1 6 e 3 x + Ce 3 x 4. y = 4 3 e 3 x + Ce 3 x 5. y = 4 3 e 3 x + Ce 3 x Explanation: The integrating factor for the first order differential equation dy dx + 3 y = 8 e 3 x is μ = e integraltext 3 dx = e 3 x . Thus after multiplying both sides by e 3 x we can rewrite it as d dx parenleftBig y e 3 x parenrightBig = 8 e 6 x . Consequently, its general solution is given by y e 3 x = integraldisplay e 6 x dx = 4 3 e 6 x + C where C is an arbitrary constant, so y = 4 3 e 3 x + C e 3 x with C an arbitrary constant. 002(part2of3)10.0points (ii) Then find the particular solution y 0 such that y 0 (0) = 8. 1. y 0 = 4 3 e 3 x + 20 3 e 3 x correct 2. y 0 = - 4 3 e 3 x + 20 3 e 3 x 3. y 0 = 4 3 e 3 x - 20 3 e 3 x 4. y 0 = 1 6 e 3 x + 20 3 e 3 x 5. y 0 = 4 3 e 3 x + 20 3 e 3 x Explanation: For the particular solution y 0 the value of C is determined by the condition y (0) = 8 since y (0) = 8 = 8 = 4 3 + C. Consequently, y 0 = 4 3 e 3 x + 20 3 e 3 x . 003(part3of3)10.0points (iii) For the particular solution y 0 in (ii), deter- mine the value of y 0 (1). 1. 4 3 e 3 + 20 3 e 3 correct 2. 1 6 e 3 - 20 3 e 3 3. 1 6 e 3 + 20 3 e 3 4. 4 3 e 3 + 20 3 e 3 5. 4 3 e 3 - 20 3 e 3
pacheco (jnp926) – Homework 6 – staron – (52840) 2 Explanation: At x = 1, therefore, y 0 (1) = 4 3 e 3 + 20 3 e 3 . 004 10.0points If y 0 is the particular solution of the differ- ential equation dy dx + 3 y + 3 e 4 x = 0 such that y (0) = 5, find the value of y 0 (1). 1. y 0 (1) = 3 7 e 4 + 32 7 e 3 2. y 0 (1) = - 3 7 e 4 + 38 7 e 3 correct 3. y 0 (1) = - 3 7 e 4 - 32 7 e 3 4. y 0 (1) = - 3 7 e 4 - 38 7 e 3 5. y 0 (1) = 3 7 e 4 + 38 7 e 3 Explanation: