Author Topic: Q1, P1--Day section  (Read 1123 times)

Victor Ivrii

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Q1, P1--Day section
« on: October 03, 2013, 07:40:13 PM »
Someone, post it! And also P2

Razak Pirani

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Re: Q1, P1--Day section
« Reply #1 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 $y$: $y=x\tan (C\ln |x|)$.
« Last Edit: October 05, 2013, 03:18:35 AM by Victor Ivrii »