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Find the values of the constants a and b for which ax + b is a particular integral of the differential equation 2y' + 5y = 10x. Hence find the general solution of 2y' + 5y = 10x .

We start by letting yp = ax+b, as suggested, and finding the derivative yp' = a. Substituing into ODE (by way of matching coefficients) gives a pair of simulataneous (algebraic) equations: (1) 2a=5b (2) -5a=...
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Answered by Ben R. • Maths tutor
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Why is 2 + 2 not equal to 12?

This might seem like an obvious question and the instinctive urge is to disregard it entirely. Perhaps you would think "well that's silly, it's just not!" Well, that's the intention of the question...
XJ
Answered by Xav J. • Maths tutor
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integrate e^x sin x dx

1/2 e x (sin x -cos x)
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Answered by Sam J. • Maths tutor
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Derive the quadratic formula (Hint: complete the square)

Firstly, the quadratic formula finds the roots of a quadratic equation. So this means f(x) = 0. A general polynomial with highest power 2 looks like: ax 2 + bx +c. Usings the two facts we just stated, we sol...
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Answered by Riu K. • Maths tutor
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How do I differentiate something of the form a^x?

You should write a x as (e lna ) x = e xlna . ln(a) is a constant, so use the chain rule to differentiate this: lna(e xlna ) = lna(a x )
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Answered by Ilya S. • Maths tutor
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