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For each of the following differential equations, find a particular solution satisfying the given condition: \frac{\mathrm{dy}}{\mathrm{dx}}=-4 \mathrm{xy}^{2}, it being given that \mathrm{y}=1 when \mathrm{x}=0
CBSE | Class 12 | Differential Equations with Variable Separable | Excercise 19A | Maths | RS Aggarwal
For each of the following differential equations, find a particular solution satisfying the given condition: \frac{\mathrm{dy}}{\mathrm{dx}}=-4 \mathrm{xy}^{2}, it being given that \mathrm{y}=1 when \mathrm{x}=0

Solution:

On rearranging the terms we obtain:
\frac{d y}{y^{2}}=-4 x d x
On integrating both sides we obtain:
\Rightarrow \int \frac{d y}{y^{2}}=-\int 4 x d x+c
\Rightarrow \frac{y^{-1}}{-1}=-\frac{4 x^{2}}{2}+c
\begin{array}{l} \Rightarrow y^{-1}=2 x^{2}+c \\ y=1 \text { when } x=0 \\ \Rightarrow(1)^{-1}=2(0)^{2}+c \\ \Rightarrow c=1 \\ \Rightarrow \frac{1}{y}=2 x^{2}+1 \\ \Rightarrow \frac{1}{2 x^{2}+1}=y \end{array}

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