By Mejlbro L.

ISBN-10: 8776812367

ISBN-13: 9788776812362

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**Sample text**

When x = 2t2 ln t − 5t + c t2 is put into the left hand side of the diﬀerential equation we get dx 2 − x = 4t ln t + 2t − 5 + 2ct − 4t ln t + 10 − 2ct = 2t + 5. dt t We have checked our solution. 5 Find a polynomial of degree 1 which is a solution of the diﬀerential equation dx + t2 x = t3 + 1, dt t ∈ R. Then ﬁnd the complete solution of the equation. A. A linear inhomogeneous diﬀerential equation of ﬁrst order. One is here invited to guess a particular solution. D. Use the hint of guessing a particular solution.

The task is to show that there to any (t, x) = (a, b) 2 exists precisely one c ∈ R, such that b = −1 + c · ea . 2 I D. Solve the equation b = −1 + c · ea with respect to c for given (a, b). com 49 Calculus 1c-1 Linear differential equation of first order 2 1 –3 –2 –1 0 2 1 3 –1 –2 Figure 19: The solution curve x = −1 through (3, −1). 2 I I. From b = −1 + c · ea we get 2 c = e−a (b + 1), which shows that the solution c exists and is uniquely determined. 2 I. Sketches of curves for x = −1 + c · et through chosen points.

C. Test. When x = 2t2 ln t − 5t + c t2 is put into the left hand side of the diﬀerential equation we get dx 2 − x = 4t ln t + 2t − 5 + 2ct − 4t ln t + 10 − 2ct = 2t + 5. dt t We have checked our solution. 5 Find a polynomial of degree 1 which is a solution of the diﬀerential equation dx + t2 x = t3 + 1, dt t ∈ R. Then ﬁnd the complete solution of the equation. A. A linear inhomogeneous diﬀerential equation of ﬁrst order. One is here invited to guess a particular solution. D. Use the hint of guessing a particular solution.

### Real functions in one variable. Simple DEs 1 by Mejlbro L.

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