Any help with this would be much appreciated (My work is in bold) :-
1a. Using the substitution u = 1+2x, or otherwise, find.
INT 4x/((1+2x)^2) dx, x>-1/2
u=1+2x
x = 1/2(u-1)
dx/du= 1/2
INT 4x/((1+2x)^2) dx = INT (4(1/2(u-1)))/u^2 du/dx du
2 INT (u-1/u^2)(1/2 du) = INT (1/u - u^-2) du
= ln |u| + 1/u +c
= ln (1+2x) + 1/(1+2x) + c.
b. Given that y= pi/4 when x = 0, solve the differential equation.
solve (1+2x)^2 dy/dx = x/sin^2y
sin^2ydy=x/(1+2x)^2 dx
INT sin^2ydy = 1/4 INT 4x/(1+2x)^2 dx.
1/2 INT (1-cos 2y) dy = 1/4(ln(1+2x)+1/(1+2x)+c)
1/2y = 1/4 sin2y = 1/4 (ln(1+2x)+1/(1+2x)+c)
2y-sin^2y = ln (1+2x) + 1/(1+2x) +c
2(pi/4) - sin^2(pi/4) = ln (1) +1/1 +c
c =0.571 (3s.f.)
Is this far correct?
2. In a chemical reaction two substances combine to form a third substance. At time t, t>_0(meant to say t more than or equal to zero, any better way of putting this on pc?), the concentration of this third substance is x and the reaction is modelled by the differential equation
dx/dt=k(1-2x)(1-4x), where k is a positive constant.
a. Solve this differential equation and hence show that
ln|(1-2x)/(1-4x)| = 2kt +c, where c is an arbitrary constant.
b. Given that x = 0 when t = 0, find an expression for x in terms of k and t.
c. Find the limiting value of the concentration x as t becomes very large.
Got no idea really how to start here.
I hope this is the right forum, I assume this comes under "Calculus".
Thanks for any help.
1a. Using the substitution u = 1+2x, or otherwise, find.
INT 4x/((1+2x)^2) dx, x>-1/2
u=1+2x
x = 1/2(u-1)
dx/du= 1/2
INT 4x/((1+2x)^2) dx = INT (4(1/2(u-1)))/u^2 du/dx du
2 INT (u-1/u^2)(1/2 du) = INT (1/u - u^-2) du
= ln |u| + 1/u +c
= ln (1+2x) + 1/(1+2x) + c.
b. Given that y= pi/4 when x = 0, solve the differential equation.
solve (1+2x)^2 dy/dx = x/sin^2y
sin^2ydy=x/(1+2x)^2 dx
INT sin^2ydy = 1/4 INT 4x/(1+2x)^2 dx.
1/2 INT (1-cos 2y) dy = 1/4(ln(1+2x)+1/(1+2x)+c)
1/2y = 1/4 sin2y = 1/4 (ln(1+2x)+1/(1+2x)+c)
2y-sin^2y = ln (1+2x) + 1/(1+2x) +c
2(pi/4) - sin^2(pi/4) = ln (1) +1/1 +c
c =0.571 (3s.f.)
Is this far correct?
2. In a chemical reaction two substances combine to form a third substance. At time t, t>_0(meant to say t more than or equal to zero, any better way of putting this on pc?), the concentration of this third substance is x and the reaction is modelled by the differential equation
dx/dt=k(1-2x)(1-4x), where k is a positive constant.
a. Solve this differential equation and hence show that
ln|(1-2x)/(1-4x)| = 2kt +c, where c is an arbitrary constant.
b. Given that x = 0 when t = 0, find an expression for x in terms of k and t.
c. Find the limiting value of the concentration x as t becomes very large.
Got no idea really how to start here.
I hope this is the right forum, I assume this comes under "Calculus".
Thanks for any help.
