3354_f05_st1

# 3354_f05_st1 - equation u = u 2 t 2 u t ANSWER tu-1 ln t =...

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Math 3354, Section 2, Sample Test # 1, Name 1. Pure Time Find the general solution of the equation u 0 = te - 2 t . ANSWER: - t/ 2 e - 2 t - e - 2 t / 4 + C 2. Exact Show that the equation u 0 = - sin( u ) - u sin( t ) 1 + t cos( u ) + cos( t ) is exact and use this fact to ﬁnd an implicit form of the general solution. ANSWER: t sin( u ) + u cos( t ) + u = C 3. Separable Use the method of separation of variables to ﬁnd the solution to the initial value problem u 0 = 1 + u 2 , u (0) = - 1. ANSWER: u = tan( t - π/ 4) 4. Separation of Variables Use separation of variables to ﬁnd the general solution of u 0 = t t 2 + 1 cos( u ) . ANSWER: u = sin - 1 ± (1 + t 2 ) 3 / 2 3 + C ² 5. Homogeneous Find the general (nonzero) implicit solution of the homogeneous
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Unformatted text preview: equation u = u 2 t 2 + u t . ANSWER: tu-1 + ln( t ) = C 6. First Order Linear Solve the ﬁrst order linear equation u =-1 t u + t . ANSWER: u = t 2 3 + C t 7. Substitution Use the substitution y = u to solve a ﬁrst order equation u 00 + u = 3 t for y . Then use the result to solve for u (note your answer contains two arbitrary constants) ANSWER: u = 3 t 2 / 2-3 t + C 1-C 2 e-t 8. Bernoulli equation Find the general solution of u =-u t + 1 tu 2 . ANSWER: u 3-1-t-3 C = 0...
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