MAT244-2013F > Quiz 1

Q1, P1--Day section

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**Victor Ivrii**:

Someone, post it! And also P2

**Razak Pirani**:

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 $y$: $y=x\tan (C\ln |x|)$.

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