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  • Need solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 21 sub question (iii)

Need solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 21 sub question (iii)

Answers (1)

Answer:

y=\frac{a}{x+a}  is the function which satisfies the differential equation

Hint:

Just differentiate the function and substitute in given equation

Given:

y=\frac{a}{x+a} 

Solution:

Differentiating on both sides with respect to x

\begin{aligned} &\frac{d y}{d x}=\frac{(x+a)(0)-a(1)}{(x+a)^{2}} \\\\ &\frac{d y}{d x}=\frac{-a}{(x+a)^{2}} \end{aligned}                                .............(i)

Substitute equation (i) in differential equation

\begin{aligned} &x \frac{d y}{d x}+y=y^{2} \\\\ &L H S=x \frac{d y}{d x}+y \end{aligned}

            \begin{aligned} &=x\left(\frac{-a}{(x+a)^{2}}\right)+y \\\\ &=\frac{-x a}{(x+a)^{2}}+\frac{a}{x+a} \\\\ &=\frac{-x a+a(x+a)}{(x+a)^{2}} \end{aligned}

            \begin{aligned} &=\frac{-x a+x a+a^{2}}{(x+a)^{2}} \\\\ &=\frac{a^{2}}{(x+a)^{2}} \\\\ &=\left(\frac{a}{(x+a)}\right)^{2} \end{aligned}

            =y^{2}                                                ..................(ii)

LHS = RHS

Thus, y=\frac{a}{x+a}  is the function which satisfies the differential equation.

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