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Solving a differential equation of the form $\\frac{dy}{dx}=x(y-a)$ using separation of variables.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Solve the differential equation \\[ \\frac{dy}{dx}=x(\\simplify{y-{a}}).\\]
", "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 $\\frac{dy}{dx}=x(\\simplify{y-{a}})$:
\n\\[ \\begin{split} &\\frac{dy}{dx}=x(\\simplify{y-{a}}) \\\\ \\implies \\frac{1}{\\simplify{y-{a}}}\\,& \\frac{dy}{dx} =x. \\end{split} \\]
\nIntegrating both sides with respect to $x$:
\n\\[ \\begin{split} \\int \\frac{1}{\\simplify{y-{a}}} \\, \\frac{dy}{dx}\\, dx &= \\int x \\, dx, \\\\ \\\\ \\implies \\int \\frac{1}{\\simplify{y-{a}}} \\,dy &= \\int 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 \\frac{1}{\\simplify{y-{a}}} \\,dy &=x \\, dx \\\\ \\ln(\\simplify{y-{a}})&\\,= \\frac{x^2}{2} +c, \\\\ \\simplify{y-{a}} &\\,=e^{\\frac{x^2}{2} +c} \\\\ y&\\,=\\simplify{e^(x^2/2+c)+{a}}. \\end{split} \\]
\n\n
Note: You can also have the answer
\n\\[y=A\\simplify{e^(x^2/2)+{a}}, \\]
\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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