Question

Resolva a equação abaixo isolando x

dy =2x✓y-1​
__
dx

Answer 1

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Equações Diferenciais

$\sf\dfrac{dy}{dx}=2x\sqrt{y-1}\\\sf\dfrac{dy}{\sqrt{y-1}}=2x~dx\\\underline{\rm integrando~dos~dois~lados~temos:}\\\displaystyle\sf\int\dfrac{dy}{\sqrt{y-1}}=\int2x~dx\\\displaystyle\sf\int\dfrac{dy}{\sqrt{y-1}}\\\underline{\rm fac_{\!\!,}a}\\\sf t=y-1\implies dt=dy\\\displaystyle\sf\int\dfrac{dy}{\sqrt{y-1}}=\int\dfrac{dt}{\sqrt{t}}=2\sqrt{t}+k\\\displaystyle\sf\int\dfrac{dy}{\sqrt{y-1}}=2\sqrt{y-1}+k$

$\displaystyle\sf\int 2x~dx=x^2+k$

$\displaystyle\sf\int\dfrac{dy}{\sqrt{y-1}}=\int 2x~dx\\\sf 2\sqrt{y-1}=x^2+k\\\sf (2\sqrt{y-1})^2=(x^2)^2+k\\\sf 4\cdot(y-1)=x^4+k\\\sf y-1=\dfrac{1}{4}x^4+k\\\huge\boxed{\boxed{\boxed{\boxed{\sf y(x)=\dfrac{1}{4}x^4+1+k}}}}\blue{\checkmark}$