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anonymous
 one year ago
Find the general solution of the equation: (x^2)(dy/dx)=(y^1/2)(3x+1)
anonymous
 one year ago
Find the general solution of the equation: (x^2)(dy/dx)=(y^1/2)(3x+1)

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Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.5we can rewrite your ODE as follows: \[\Large \begin{gathered} {x^2}y' = \left( {3x + 1} \right)\sqrt y \hfill \\ \hfill \\ \frac{{dy}}{{\sqrt y }} = \frac{{3x + 1}}{{{x^2}}}dx \hfill \\ \end{gathered} \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.5now we can easily integrate both sides, so we get: \[\Large 2\sqrt y = 3\ln \left x \right  \frac{1}{x} + k\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.5finally,dividing ooth sides by 2, we get: \[\Large \sqrt y = 3\ln \left( {\sqrt x } \right)  \frac{1}{{2x}} + k\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.5squaring both sides, we get: \[\Large y\left( x \right) = {\left( {3\ln \left( {\sqrt x } \right)  \frac{1}{{2x}} + k} \right)^2}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.5where, as usual, k is the arbitrary real constant of integration

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Are there any more steps to that?
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