Math 225 Differential Equations Notes Chapter 2
Math 225 Differential Equations Notes Chapter 2
Math 225 Differential Equations Notes Chapter 2
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Example 6: Solve<br />
(2xy − sec 2 x)dx + (x 2 + 2y)dy = 0<br />
Solution:<br />
Here M(x, y) = (2xy − sec 2 x) and N(x, y) = (x 2 + 2y) Testing for<br />
exactness:<br />
∂M ∂N<br />
= 2x =<br />
∂y ∂x<br />
Integrate M(x, y) with respect to x<br />
F (x, y) =<br />
∫<br />
(2xy − sec 2 x)dx + g(y) = x 2 y − tan(x) + g(y)<br />
Next, take the partial derivative w.r.t y of the integral equation substituting<br />
∂F<br />
∂y = N(x, y) = (x2 + 2y)<br />
x 2 + 2y = x 2 + g ′ (y)<br />
so g ′ (y) = 2y implying g(y) = y 2 substituting in the equation for<br />
F (x, y) we get<br />
F (x, y) = x 2 y − tan(x) + g(y) = x 2 y − tan(x) + y 2<br />
and the implicit solution of the DE is<br />
x 2 y − tan(x) + y 2 = C<br />
18
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