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Solving a differential equation of the form $\\frac{dy}{dx}=axy$ using separation of variables.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Solve the differential equation \\[ \\frac{dy}{dx}=\\var{a}xy.\\]
", "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 $\\frac{dy}{dx}=\\var{a}xy$:
\n\\[ \\begin{split} &\\frac{dy}{dx}=\\var{a}xy \\\\\\\\ \\implies \\frac{1}{y}& \\frac{dy}{dx} = \\var{a}x. \\end{split} \\]
\nIntegrating both sides with respect to $x$:
\n\\[ \\begin{split} & \\int \\frac{1}{y} \\frac{dy}{dx} dx = \\int \\var{a}x \\, dx, \\\\ \\\\ \\implies &\\int \\frac{1}{y} 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}&\\quad \\int &\\frac{1}{y} dy &\\,= \\int \\var{a} x \\, dx \\\\ &\\implies &\\ln(y) &\\,= \\simplify{{a}x^2/2} + c, \\\\ &\\implies &\\quad y &\\,= e^{\\simplify{{a}x^2/2+c}}. \\end{split} \\]
\n\nNote: You can also have the answer
\n\\[ y =Ae^{\\simplify{{a/2}x^2}}, \\]
\nwhere $A=e^c$. This shows there is still a constant, but indicates it is different to the original constant of integration, $c$.
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