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Ilia_Sergeevich [38]
3 years ago
6

Solve the given initial-value problem. (x + y)^2 dx + (2xy + x^2 - 8) dy = 0, y(1) = 1

Mathematics
1 answer:
erica [24]3 years ago
7 0

Notice that

\dfrac{\partial(x+y)^2}{\partial y}=2(x+y)=2x+2y

\dfrac{\partial(2xy+x^2-8)}{\partial x}=2y+2x

so the ODE is exact, and we can look for a solution of the form f(x,y)=C. Differentiating both sides gives

\dfrac{\partial f}{\partial x}\,\mathrm dx+\dfrac{\partial f}{\partial y}\,\mathrm dy=0

\dfrac{\partial f}{\partial x}=(x+y)^2=x^2+2xy+y^2\implies f(x,y)=\dfrac{x^3}3+x^2y+xy^2+g(y)

\dfrac{\partial f}{\partial y}=2xy+x^2-9=x^2+2xy+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=-9\implies g(y)=-9y+C

\implies f(x,y)=\dfrac{x^3}3+x^2y+xy^2-9y+C=C

so that the solution to the ODE is

\dfrac{x^3}3+x^2y+xy^2-9y=C

Given that y(1)=1, we have

\dfrac13+1+1-9=C\implies C=-\dfrac{20}3

so that the particular solution is

\boxed{\dfrac{x^3}3+x^2y+xy^2-9y=-\dfrac{20}3}

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