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  • Need solution for RD Sharma Maths Class 12 Chapter 21 Differential Equation exercise 21.5 question 17.

Need solution for RD Sharma Maths Class 12 Chapter 21 Differential Equation exercise 21.5 question 17.

Answers (1)

Answer: y=\frac{2}{3}(a+x)^{\frac{3}{2}}-2a(a+x)^{\frac{1}{2}}+C

Hint: You have to integrate by applying integration of xn.

Given: \sqrt{a+x} \; dy+x \: dx=0

Solution:  \frac{dy}{dx}=f(x)

\begin{aligned} &\sqrt{a+x} d y=-x d x \\ &d y=\frac{-x}{\sqrt{a+x}} d x \end{aligned}

Integrate on both sides

\begin{aligned} &\int d y=\int \frac{-x}{\sqrt{a+x}} d x+C \quad[a+x=t, d x=d t, x=t-a] \\ &\int d y=\int \frac{t-a}{\sqrt{t}} d t+C \\ &\int d y=\int\left(\frac{t}{\sqrt{t}}-\frac{a}{\sqrt{t}}\right) d t \\ &\int d y=\int\left(\sqrt{t}-a \cdot t^{-\frac{1}{2}}\right) d t \end{aligned}

y=\frac{t^{\frac{8}{2}}}{\frac{3}{2}}-a \cdot \frac{t^{\frac{1}{2}}}{\frac{2}{2}}+C

y=\frac{2}{3}(a+x)^{\frac{3}{2}}-2a(a+x)^{\frac{1}{2}}+C

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