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For each of the following differential equations, find a particular solution satisfying the given condition : x d y=\left(2 x^{2}+1\right) d x(x \neq 0), given thaty =1 when x=1
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 : x d y=\left(2 x^{2}+1\right) d x(x \neq 0), given thaty =1 when x=1

Solution:

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

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