MYTUTOR SUBJECT ANSWERS

489 views

Show that substituting y = xv, where v is a function of x, in the differential equation "xy(dy/dx) + y^2 − 2x^2 = 0" (with x is not equal to 0) leads to the differential equation "xv(dv/dx) + 2v^2 − 2 = 0"

This is the first part of a Step 1 question (2012 question 8), and is fairly typical in that it requires A Level Maths understanding to be applied in a number of different ways, out of the usual context.

We begin by finding a new expression for (dy/dx), in terms of v. We do this by differentiating "y = xv", and getting:

dy/dx = x(dv/dx) + v

By substituting this and "y = xv" into the original differential equation, we get:

(x)(xv)(x(dv/dx) + v) + (xv)^2 - 2(x^2) = 0

(x^3)v(dv/dx) + 2(xv)^2 - 2(x^2) = 0

We can divide by x^2, since we know that x does not equal zero. This gives the required result:

xv(dv/dx) + 2(v^2) - 2 = 0

The rest of the question first asks you to find a solution to the original equation. The new equation is now separable, and can be solved to find v. This can be substituted back into "y = xv", giving a solution for y. Try it if you like. It should look something like:

(x^2)(y^2 - x^2) = C (where C is a constant)

The question then asks you to solve (again with x is not equal to zero):

y(dy/dx) + 6x + 5y = 0

This can be done with the same substitution, which again makes it separable. Using partial fractions, integrating, finding v in terms of x (or vice versa), and substituting allows a solution to be found. This should (although it is still pretty ugly, particularly on a computer screen) look like:

((y + 3x)^3)/((y + 2x)^2) = B (where B is a constant)

Will W. GCSE Maths tutor, A Level Maths tutor, A Level Further Mathem...

10 months ago

Answered by Will, an Uni Admissions Test .STEP. tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

31 SUBJECT SPECIALISTS

£36 /hr

Joe B.

Degree: Mathematics G100 (Bachelors) - Bath University

Subjects offered:.STEP., Maths+ 3 more

.STEP.
Maths
Further Mathematics
.MAT.
-Personal Statements-

“Hi! I'm Joe, a friendly, experienced and patient tutor with in-depth knowledge of both the old and new A Level Maths & Further Maths specifications.”

£28 /hr

Henri F.

Degree: Aerospace Engineering PhD Spacecraft Control (Doctorate) - Bristol University

Subjects offered:.STEP., Physics+ 4 more

.STEP.
Physics
Maths
Further Mathematics
Extended Project Qualification
.PAT.

“Aerospace Engineering PhD candidate in spacecraft control with 7 years of experience in tutoring.”

£25 /hr

Lewis C.

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered:.STEP., Maths+ 3 more

.STEP.
Maths
Further Mathematics
.MAT.
-Oxbridge Preparation-

“Friendly first-year Mathematics student studying at Trinity College, Cambridge. Two years tutoring experience.”

MyTutor guarantee

|  7 completed tutorials

About the author

£30 /hr

Will W.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered:.STEP., Physics+ 2 more

.STEP.
Physics
Maths
Further Mathematics

“Hi, I'm Will. I study Maths and Physics and really enjoy talking about them. I am patient and caring, so will take the time to refine a student’s understanding.”

You may also like...

Posts by Will

How do you calculate the derivative of cos inverse x?

How do you integrate (sinx)^2?

How much work must be done on a 4.0kg frictionless trolley, to accelerate it from rest to a velocity of 5.0m/s?

Show that substituting y = xv, where v is a function of x, in the differential equation "xy(dy/dx) + y^2 − 2x^2 = 0" (with x is not equal to 0) leads to the differential equation "xv(dv/dx) + 2v^2 − 2 = 0"

Other Uni Admissions Test .STEP. questions

Prove that any number of the form pq, where p and q are prime numbers greater than 2, can be written as the difference of two squares in exactly two distinct ways.

Show that substituting y = xv, where v is a function of x, in the differential equation "xy(dy/dx) + y^2 − 2x^2 = 0" (with x is not equal to 0) leads to the differential equation "xv(dv/dx) + 2v^2 − 2 = 0"

Show that if a polynomial with integer coefficients has a rational root, then the rational root must be an integer. Hence, show that x^n-5x+7=0 has no rational roots.

How can I integrate e^x sin(x)?

View Uni Admissions Test .STEP. tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok