A) \[kx={{e}^{-y/x}}\]
B) \[kx={{e}^{y/x}}\]
C) \[kx={{e}^{x/y}}\]
D) \[kx={{e}^{-x/y}}\]
Correct Answer: A
Solution :
Idea This is homogeneous differential equation Form \[\frac{dy}{dx}=\]f (x,y}. This equation is solved by putting y = vx, where v = v (x), a function of x. Given that \[({{x}^{2}}-2xy)dy=({{x}^{2}}-3xy+2{{y}^{2}})dx=0\] Put\[y=vx,\frac{dy}{dx}=v+x\frac{dv}{dx}\] \[x\frac{dv}{dx}=\frac{1-3v+2{{v}^{2}}}{2v-1}-v=\frac{1-2v}{2v-1}\] \[\Rightarrow \] \[x\frac{dv}{dx}=-1\] \[-dv=\frac{dx}{x}\] \[-\int_{{}}^{{}}{dv=}\int_{{}}^{{}}{\frac{1}{x}}dx\] \[-v=\log x+\log k\] \[-v=\log kx\] \[kx={{e}^{-v}}={{e}^{-y/x}}\] TEST Edge Equation reducible to the separable type, to homogeneous form based questions are asked. To solve such type of question, students are advised to understand the concept of differential equation.
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