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anonymous
 4 years ago
How would I start working a first order differential equation (with an initial value) like this?
(x^3  y) dx + x dy = 0, y(1) = 3
anonymous
 4 years ago
How would I start working a first order differential equation (with an initial value) like this? (x^3  y) dx + x dy = 0, y(1) = 3

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0The given equation is : \[x ^{3} + x*dy(x)/dx  y(x) = 0\] = \[dy(x)/dx  y(x)/x = x ^{2}\] Now put, \[\mu(x) = e ^{\int\limits 1/x dx} = 1/x\] Multiply both sides of the equation by \[\mu(x)\]to get : \[[dy(x)/dx]/x  y(x)/x ^{2} = x\] = \[[dy(x)/dx]/x +d/dx(1/x)y(x) = x\] Now apply the reverse product rule to get: \[d/dx(y(x)/x) = x\] or,\[\int\limits d/dx(y(x)/x) dx = \int\limits x dx\] or,\[y(x)/x = x ^{2}/2 +c _{1}\] Now you can proceed easily!
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