Form the differential equation from the following primitives wher… - Vidyadip

Question

Form the differential equation from the following primitives where constants are arbitrart:$\text{y}=\text{cx}+2\text{c}^2+\text{c}$

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Answer

The equation of family of curves is $\text{y}=\text{cx}+2\text{c}^2+\text{c}\ ...(1)$ where c is an arbitrary constant. This equation contains only one arbitrary constant, so we shall get a differential equation of first order. Differentiating equation (1) with respect to x, we get $\frac{\text{dy}}{\text{dx}}=\text{c}\ ...(2)$ $\Rightarrow\frac{\text{y}}{2}\frac{\text{dy}}{\text{dx}}=\text{a}\ ...(2)$ Putting the value of a in equation (1), we get $\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+2\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\Big(\frac{\text{dy}}{\text{dx}}\Big)^3$It is the required differential equation.

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