From the differential equation of the family of circles having centre on y-axis and radius 3 units.
From the differential equation of the family of circles having centre on y-axis and radius 3 units.
Solution and Explanation
Let the centre of the circle on y-axis be(0,b).
The differential equation of the family of circles with centre at (0,b)and radius 3 is as
follows:
x2+(y-b)2=32
\(\Rightarrow \) x2+(y-b)2=9 ...(1)
Differentiating equation(1) with respect to x, we get:
2x+2(y-b).y'=0
\(\Rightarrow\) (y-b).y'=-x
\(\Rightarrow\) y-b=-\(\frac{x}{y'}\)
Substituting the value of (y-b)in equation(1),we get:
\(x^2+(-\frac{x}{y'})^2=9\)
\(\Rightarrow x^2[1+\frac{1}{(y')^2}]=9\)
\(\Rightarrow\) x2((y')2+1)=9(y')2
\(\Rightarrow\) (x2-9)(y')2+x2=0
This is the required differential equation.
Top Questions on Differential equations
- If \[\frac{dx}{dy} = \frac{1 + x - y^2}{y}, \quad x(1) = 1,\]then $5x(2)$ is equal to:
- JEE Main - 2024
- Mathematics
- Differential equations
- Let $\alpha$ be a non-zero real number. Suppose $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(0) = 2$ and \[\lim_{x \to \infty} f(x) = 1.\]If $f'(x) = \alpha f(x) + 3$, for all $x \in \mathbb{R}$, then $f(-\log 2)$ is equal to ________ .
- JEE Main - 2024
- Mathematics
- Differential equations
- Let\(f(x) = |2x^2 + 5|x - 3|, x \in \mathbb{R}\). If \(m\) and \(n\) denote the number of points were \(f\)is not continuous and not differentiable respectively, then\(m + n\)is equal to:
- JEE Main - 2024
- Mathematics
- Differential equations
- Let \( y = y(x) \) be the solution of the differential equation \[\left( 1 + x^2 \right) \frac{dy}{dx} + y = e^{\tan^{-1}x}, \, y(1) = 0.\]Then \( y(0) \) is:
- JEE Main - 2024
- Mathematics
- Differential equations
- Let \( y = y(x) \) be the solution of the differential equation \[\left( 2x \log_e x \right) \frac{dy}{dx} + 2y = \frac{3}{x} \log_e x, \, x>0 \, \text{and} \, y(e^{-1}) = 0.\] Then, \( y(e) \) is equal to:
- JEE Main - 2024
- Mathematics
- Differential equations
Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
- For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
Concepts Used:
General Solutions to Differential Equations
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
For example,
Read More: Formation of a Differential Equation