First slide

Methods of solving first order first degree differential equations

Question

$The solution of the D.E {yy}^{1} = x (\frac{y^{2}}{x^{2}} + \frac{f (\frac{y^{2}}{x^{2}})}{f^{1} (\frac{y^{2}}{x^{2}})}) is$

Moderate

A

$f (y^{2} {/x}^{2}) = {cx}^{2}$
B

$x^{2} f (y^{2} {/x}^{2}) = c^{2} y^{2}$
C

$x^{2} f (y^{2} {/x}^{2}) = c$
D

$f (y^{2} {/x}^{2}) = cy/x$

Solution

$\begin{array}{l} \frac{y}{x} \frac{dy}{dx} = \frac{y^{2}}{x^{2}} + \frac{f (\frac{y^{2}}{x^{2}})}{f^{1} (\frac{y^{2}}{x^{2}})} \\ Put y = vx \\ \frac{dy}{dx} = v + x \frac{dv}{dx} \\ \to v (v + x \frac{dv}{dx}) = v^{2} + \frac{f (v^{2})}{f^{1} (v^{2})} \\ \to vx \frac{dv}{dx} = \frac{f (v^{2})}{f^{1} (v^{2})} \\ \int \frac{f^{1} (v^{2}) 2 v d v}{f (v^{2})} = 2 \int \frac{d x}{x} \\ \to \ln [f (v^{2})] = \ln x^{2} + \ln c \\ \to f (v^{2}) = {cx}^{2} \\ \to f (\frac{y^{2}}{x^{2}}) = {cx}^{2} \end{array}$

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