The solution of the differential equation `y_1y_3=y_2^2` is a. `x=C_1e^(C_2 \ y)+C_3` b. `y=C_1e^(C_(2) \ x)+C_3` c. `2x=C_1e^(C_(2) \ y)+C_3` d. none of these

Explanation

`y_{1} y_{3}=y_{2}^{2}`

`frac{y_{3}}{y_{2}}=frac{y_{2}}{y_{1}}`

`Rightarrow frac{(frac{d^{3} y}{d x^{3}})}{(frac{d^{2} y}{d x^{2}})}=frac{(frac{d^{2} y}{d x^{2}})}{(frac{d y}{d x})}`

`Rightarrow int frac{frac{d}{d x}(frac{d^{2} y}{d x^{2}})}{(frac{d^{2} y}{d x^{2}})}=int frac{frac{d}{d x}(frac{d y}{d x})}{(frac{d y}{d x})}`

`Rightarrow ln (frac{d^{2} y}{d x^{2}})=ln (frac{d y}{d x})+ln C_{4}`

`Rightarrow frac{d^{2} y}{d x^{2}}=C_{4} frac{d y}{d x}`

`Rightarrow int frac{frac{d}{d x}(frac{d y}{d x})}{(frac{d y}{d x})}=int C_{4}`

`ln (frac{d y}{d x})=C_{4} x+C_{5}`

...