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Find the general solution of the differential equation: d^2x/dt^2 + 5dx/dt + 6x = 2cos(t) - sin(t)

First solve complementary function, i.e. d 2 x/dt 2 + 5dx/dt + 6x = 0. To do so, let x = e mt , where m = arbitrary constant. Differentiating gives dx/dt = m e mt and d 2 x/dt 2 = m 2 e mt . Substituting int...
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Answered by Mick G. • Maths tutor
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Using the product rule, differentiate: y = (x^2 - 1)(x^3 + 3).

y=(x 2 -1)(x 3 +3) let u=x 2 -1 u'=2x let v=x 3 +3 v'=3x 2 uv'+u'v=(x 2 -1)(3x 2 )+(2x)(x 3 +3) =(3x 4 -3x 2 )+(2x 4 +6X) = 5x 4 -3x 2 +6x
Answered by • Maths tutor
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Describe the causes of the First World War

There were 4 main causes leading to the First World War. Firstly, the immediate cause of the war was the assassination of Archduke Franz Ferdinand in 1914 by the Black Hand organisation of Serbian nationalis...
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Answered by Aiden S. • History tutor
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Translate the following: If it hadn't rained, she would have gone to the park.

S'il n'avait pas plu, elle serait allée au parc. Just as in the English example, this 'Si' clause requires the pluperfect tense and the conditional perfect. 'Si' clauses are constructed around two essential ...
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Answered by Milly L. • French tutor
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integral of (tan(x))dx using the substitution u = cos(x)

given u = cos(x), therefore du/dx=-sin(x), as tan(x)=sin(x)/cos(x), can rewrite tan(x)=(-du/dx)/u, therefore integral can become [(-1/u)du], after inegrating you are left with -ln(u)+c, therefore ln(1/u)+c, ...
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Answered by Frederick R. • Maths tutor
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