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Solving a differential equation of the form $\\frac{dy}{dx}=axy^2$ using separation of variables.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Solve the differential equation \\[ \\frac{dy}{dx}=\\var{a}xy^2.\\]
", "advice": "If we have a differential equation of the form \\[ \\frac{dy}{dx} = f(x) g(y),\\] we are able to find a solution to this equation using the method Separation of Variables.
\nWe can rewrite the above equation in the form \\[ \\frac{1}{g(y)} \\frac{dy}{dx} = f(x).\\]
\nIf we then integrate with respect to $x$:
\n\\[ \\begin{split} &\\int \\frac{1}{g(y)} &\\frac{dy}{dx} dx = \\int f(x) dx, \\\\\\\\ \\implies &\\int \\frac{1}{g(y)} &dy = \\int f(x) dx. \\end{split} \\]
\nIntegrating both sides, we can write $y$ as a function of $x$.
\nFollowing this method for $\\frac{dy}{dx}=\\var{a}xy^2$:
\n\\[ \\begin{split} &\\frac{dy}{dx}=\\var{a}xy^2 \\\\\\\\ \\implies \\frac{1}{y^2}& \\frac{dy}{dx} = \\var{a}x\\\\\\\\ \\implies y^{-2}& \\frac{dy}{dx} = \\var{a}x. \\end{split} \\]
\nIntegrating both sides with respect to $x$:
\n\\[ \\begin{split} & \\int y^{-2} \\frac{dy}{dx} dx = \\int \\var{a}x \\, dx, \\\\ \\\\ \\implies &\\int y^{-2} dy = \\int \\var{a} x \\, dx. \\end{split} \\]
\nTaking the integral of both sides, we are able to find a solution to the differential equation:
\n\\[ \\begin{split} \\int y^{-2} dy &\\,= \\int \\var{a} x \\, dx, \\\\ -y^{-1}&\\,= \\simplify{{a/2}x^2 +c}, \\\\ -y&\\,= \\frac{1}{\\simplify{{a/2} x^2 +c}}, \\\\ y&\\,= -\\frac{1}{\\simplify{{a/2} x^2 +c}}. \\end{split} \\]
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