# Toronto Math Forum

## MAT244-2013F => MAT244 Math--Tests => Quiz 1 => Topic started by: Victor Ivrii on October 03, 2013, 07:40:13 PM

Title: Q1, P1--Day section
Post by: Victor Ivrii on October 03, 2013, 07:40:13 PM
Someone, post it! And also P2
Title: Re: Q1, P1--Day section
Post by: Razak Pirani on October 04, 2013, 09:34:54 PM
Problem 1: Section 2.2, #31

Solve the homogeneous differential equation using the substitution $y(x) = xv(x)$
$$dy/dx = (x^2 + xy + y^2)/x^2$$
First note: Since $y(x) = xv(x)$,  $dy/dx = v(x) + x(dv/dx)$. Dividing the numerator and denominator by $x^2$ and substituting $v = y/x$ yields the homogeneous equation
$$v + x(dv/dx) = 1 + v + v^2\implies dx/x = dv/(1 + v^2).$$
Take the integral of both sides
$$\ln|x| + C = \arctan(v).$$
Substitute $y/x = v$
$$\arctan(y/x) - \ln|x| = C.$$

Observe how I modified the source to provide a proper formatting. Note that the last equation could be resolved with respect to $y$: $y=x\tan (C\ln |x|)$.