Separable

STEP 2007, Math II, #6: (i) Differentiate and and simplify your answers.

Hence find

(ii) Find the two solutions of the differential equation

STEP 2002, Math III, #6: Find all the solution curves of the differential equation

that pass through either of the points

(i)

(ii)

Show also that and are solutions of the differential equation. Sketch all these solution curves on a single set of axes.

STEP 1998, Math I, #8: Fluid flows steadily under a constant pressure gradient along a straight tube of circular cross-section

of radius

. The velocity of a particle of the fluid is parallel to the axis of the tube and depends only on the distance from

the axis. The equation satisfied by

is

where

is constant. Find the general solution for .

when . Find in terms of , and .

The volume

flowing through the tube per unit time is given by

.

Find

.

Step 2001, Math I, #7: In a cosmological model, the radius

of the universe is a function of the age of the universe.

The function

satisfies the three conditions:

, for , for (*)

where

denotes the second derivative of . The function is defined by

(i) Sketch a graph of . By considering a tangent to the graph, show that .

(ii) Observations reveal that

, where is a constant. Derive an expression for . What range of values of is

consistent with the three conditions (*) ?

(iii) Suppose, instead, that observations reveal that

, where is constant. Show that this is not consistent with

conditions (*) for any value of

satisfies and for all .

(i) Give an example of such a function.

(ii) The function

satisfies

and Show that , for any positive integer .

(iii) Let

be the solution of the differential equation

when . Show that as , where .

STEP 2005, Math II, #8: For

the curve C is defined by

with when . Show that

and hence that for large positive

Draw a sketch of C.

On a separate diagram draw a sketch of the two curves defined for

by

be the solution of the differential equation

that satisfies the condition when . Find in terms of and show that as .

(ii) Let

be any solution of the differential equation

such that, as , tends to a finite non-zero limit, which you should determine.

STEP 2005, Math III, #2: Find the general solution of the differential equation

, and show that it can be written in the form , where is an arbitrary constant. Sketch this curve.

Find an expression for

and show that

(i) Show that, if , the points on the curve whose distance from the origin is

least are

(ii) If , determine the points on the curve whose distance from the origin is least.

STEP 2002, Math II, #8: Find

and for the solution curve to be continuous (that is, for there to be

no "jump" in the value of

) at .

Solve the differential equation

given that when and that is continuous at . Write down the following limits:

(i) ; (ii)

STEP 2004, Math II, #8: Let

satisfy the differential equation

and the condition

and sketch the graph of the solution for .

for , using the same axes as your previous sketch.

By setting , solve the equation in this case.

(iii) Use the result (which you need not prove)

for ,

to sketch, without solving the equation, the graph of the solution of the equation in the case

at a value of less than .

STEP 2003, Math II, #8: It is given that

satisfies

for for each of these cases.

STEP 2003, Math I, #8: A liquid of fixed volume

is and the volume of B at time is where and depend on and

. The rate at which A converts into B is given by , where is a positive constant. Show that if both and are strictly positive at the start, then at time

where

is a constant.

Does A ever completely convert to B? Justify your answer.

And some Non-separable

STEP 2008, Math II, #7: (i) By writing , where is a function of , find the solution of the equation

for which when .

(ii) find the solution of the equation

for which

when .

(iii) Give, without proof, a conjecture for the solution of the equation

for which

denotes .

(i) Given that

show by differentiating with respect to

that

Hence show that

where

in terms of if when .

(ii) Given instead that

when , show that

and find

in terms of .

STEP 2004, Math III, #8: Show that if

then the substitution gives a linear differential equation for .

Hence or otherwise solve the differential equation

Determine the solution curves of this equation which pass through , and and sketch graphs of all three

curves on the same axes.

STEP 2011, Math I, #7: In this question, you may assume that

is . The initial height of the water is and water flows into the tank at a constant

rate. The cross-sectional area of the tank is constant.

(i) Suppose that water leaks out at a rate proportional to the height of the water in the tank, and that when the height reaches ,

where

is a constant greater than 1, the height remains constant. Show that

,

for some positive constant

. Deduce that the time taken for the water to reach height is given by

and that for large values of .

(ii) Suppose that the rate at which water leaks out of the tank is proportional to (instead of ), and that when the height

reaches

, where is a constant greater than 1, the height remains constant. Show that the time taken for the water

to reach height

is given by

, and that for large values of .