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Solve the differential equation dy/dx = y/x(x + 1) , given that when x = 1, y = 1. Your answer should express y explicitly in terms of x.

Rearrange differential equation to get 1/x(x+1) dx = 1/y dy. Separate x side into partial fractions where 1/x(x+1) = 1/x - 1/(x+1). Integrate each side. Resulting equation involves natural logs. Substitut...

Answered by Alexander T. • Maths tutor

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The expansion of (1+x)^4 is 1 + 4x +nx^2 + 4x^3 + x^4. Find the value of n. Hence Find the integral of (1+√y)^4 between the values 1 and 0 (one top, zero bottom).

Using Binomial expansion or Pascal's triangle, expand (1+x)^4 to get 1+4x+6x^2+4x^3+x^4. Then, by substituting √y for x, get 1 + 4y^1/2 + 6y +4y^3/2 +y^2. Then, using the rules of integration, the expansi...

Answered by Tutor41123 D. • Maths tutor

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Express 4 sin(x) – 8 cos(x) in the form R sin(x-a), where R and a are constants, R >0 and 0< a< π/2

4 sin(x) – 8 cos(x)= Rsin(x-a) here use double angle formula

4 sin(x) – 8 cos(x)= Rsin(x)cos(a)-Rcos(x)sin(a) Rearrange so in same format as LHS

4 sin(x) – 8 cos(x)= Rcos(a)sin(x)-Rsin(a)cos...

Answered by Simon E. • Maths tutor

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The line L has equation y = 5 - 2x. (a) Show that the point P (3, -1) lies on L. (b) Find an equation of the line perpendicular to L that passes through P.

(a) To confirm that point P lies on L, we must substitute x = 3 into the equation and see if we get y = -1. y = 5 - 2(3) = -1, therefore P lies on the line L (b) The gradient of the perpendicular line is ...

Answered by Kitty S. • Maths tutor

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