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luda_lava [24]
2 years ago
12

If dy/dx equals xy squared and if y = 1 when x = 0, then when y = 3, x is equal to?

Mathematics
2 answers:
Ymorist [56]2 years ago
5 0
\displaystyle
\dfrac{dy}{dx}=xy^2\\
dy=xy^2 \, dx\\
\dfrac{1}{y^2}\, dy=x\, dx\\
\int \dfrac{1}{y^2}\, dy=\int x\, dx\\
-\dfrac{1}{y}=\dfrac{x^2}{2}+C\\\\
-\dfrac{1}{1}=\dfrac{0^2}{2}+C\\
C=-1\\\\-\dfrac{1}{3}=\dfrac{x^2}{2}-1\\
-2=3x^2-6\\
3x^2=4\\
x^2=\dfrac{4}{3}\\
x=\sqrt{\dfrac{4}{3}} \vee x=-\sqrt{\dfrac{4}{3}}\\
x=\dfrac{2}{\sqrt3} \vee x=-\dfrac{2}{\sqrt3}\\
x=\dfrac{2\sqrt3}{3} \vee x=-\dfrac{2\sqrt3}{3}




S_A_V [24]2 years ago
4 0
This is a separable differential equation, so let's start of there. Let's separate the variables to their own side with the respective differentials:
\frac{dy}{dx} = xy^2
dy = (xy^2) dx
\frac{1}{y^2} dy = x dx

Let's integrate both sides (it's separable, so we can do this):
\int\ { \frac{1}{y^2} } \, dy =  \int\ {x} \, dx
- \frac{1}{y} =  \frac{x^2}{2} + C

Now, let's plug in the values we are given to find the constant "C":
- \frac{1}{1}  =\frac{0^2}{2}+C
-1 = C

Let's rewrite the equation, with C in it, then solve for x because we need to ultimately find x:
- \frac{1}{y}  = \frac{x^2}{2} - 1
x =  \sqrt{2(- \frac{1}{y}+1)}

Let's plug in y = 3 and solve for x:
x = \sqrt{2(- \frac{1}{3}+1)} = \sqrt{ 2( \frac{2}{3}) } = \sqrt{ \frac{4}{3} }

Let's simplify and rationalize the denominator:
x =  \sqrt{ \frac{4}{3}} = 2 \sqrt{ \frac{1}{3}} = 2  \frac{ \sqrt{3} }{3}

So, your answer is D.

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