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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 17 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 17 Maths Textbook Solution.

Answers (1)

Answer: 3y^{{}'}\left ( y^{{}'''} \right )^{2}=y^{{}'''}\left ( 1+\left ( y^{{}'} \right )^{2} \right )

Hint: The number of constant is equal to the number of times we need to  differentiate.

Given: x^{2}+y^{2}+2ax+2by+C=0, find differential equation, not containing constants.

Solution: x^{2}+y^{2}+2ax+2by+C=0

Differentiate w.r.t x,

2x + 2yy^{{}'} + 2a + 2by^{{}'} = 0

Again, differentiate w.r.t x

2 + 2(y^{{}'})^{2} + 2yy^{{}''} + 2by^{{}''} = 0

1 + (y^{{}'})^{2} + yy^{{}''} + by^{{}''} = 0..taking 2 common

b=\frac{-\left (1+\left ( y^{{}'} \right )^{2}+yy^{{}''} \right )}{y^{{}''}}

We have,

1+\left(y^{{}'}\right)^{2}+y y^{{}''}+b y^{{}''}=0

Again differentiate,

2 y^{{}'} y^{{}''}+y^{{}'} y^{{}''}+y y^{{}'''}+b y^{{}'''}=0

On substituting values of (b),

\begin{aligned} &3 y^{{}'} y^{{}''}+y y^{{}'''}+\left(\frac{-\left(1+\left(y^{{}'}\right)^{2}+y y^{{}''}\right.}{y^{{}''}}\right) y^{{}'''}=0 \\ &3 y^{{}'}\left(y^{{}''}\right)^{2}+y y^{{}''} y^{{}'''}-y^{{}''}-\left(y^{{}'}\right)^{2} y^{{}'''}-y y^{{}'''} y^{{}''}=0 \\ &3 y^{{}'}\left(y^{{}''}\right)^{2}=y^{{}'''}\left(1+\left(y^{{}'}\right)^{2}\right) \end{aligned}

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