A) \[f({{y}^{2}}/{{x}^{2}})=c{{x}^{2}}\]
B) \[{{x}^{2}}f({{y}^{2}}/{{x}^{2}})={{c}^{2}}{{y}^{2}}\]
C) \[{{x}^{2}}f({{y}^{2}}/{{x}^{2}})=c\]
D) \[f({{y}^{2}}/{{x}^{2}})=cy/x\]
Correct Answer: A
Solution :
| [a] The given equation can be written as |
| \[\frac{y}{x}\frac{dy}{dx}=\left\{ \frac{{{y}^{2}}}{{{x}^{2}}}+\frac{f({{y}^{2}}/{{x}^{2}})}{f'({{y}^{2}}/{{x}^{2}})} \right\}\] |
| The above equation is a homogeneous equation. |
| Putting \[y=vx,\]we get |
| \[v\left[ v+x\frac{dv}{dx} \right]={{v}^{2}}+\frac{f({{v}^{2}})}{f'({{v}^{2}})}\] |
| or \[vx\frac{dv}{dx}=\frac{f({{v}^{2}})}{f'({{v}^{2}})}\] (variable separable) |
| or \[\frac{2vf'({{v}^{2}})}{f({{v}^{2}})}dv=2\frac{dx}{x}\] |
| Now, integrating both sides, we get |
| \[\log f({{v}^{2}})=log{{x}^{2}}+\log c[logc=constant]\] |
| or \[\log f({{v}^{2}})=logc{{x}^{2}}orf({{v}^{2}})=c{{x}^{2}}\] or |
| \[f({{y}^{2}}/{{x}^{2}})=c{{x}^{2}}\] |
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