Differential Equations

Mediummathematics

The differential equation of the family of parabolas $y^{2} = 4 a (x + a)$ is:

Select the correct option:

$y (y^{'})^{2} + 2 x y^{'} - y = 0$

$y (y^{'})^{2} + 2 x y^{'} + y = 0$

$y^{'} + y = 0$

$y^{''} = 0$

🔒 Solution Hidden from View

Submit your answer to unlock the detailed step-by-step solution.

About This Question

Subject: mathematics
Chapter: differential equations
Topic: differential equations
Difficulty: Medium
Year: 2025

Solution

Correct Answer:

$y (y^{'})^{2} + 2 x y^{'} - y = 0$

Start with the equation: $y^{2} = 4 a x + 4 a^{2}$ .
Differentiate with respect to $x$ : $2 y y^{'} = 4 a ⟹ a = \frac{y y ^{'}}{2}$ .
Eliminate parameter ' $a$ ' by substituting back: $y^{2} = 4 (\frac{y y ^{'}}{2}) x + 4 (\frac{y y ^{'}}{2})^{2}$ $y^{2} = 2 x y y^{'} + y^{2} (y^{'})^{2}$ .
Simplify: Since parabolas are non-degenerate, $y$ is not identically 0. Divide by $y$ : $y = 2 x y^{'} + y (y^{'})^{2}$ $y (y^{'})^{2} + 2 x y^{'} - y = 0$ .

This medium difficulty mathematics question is from the chapter differential equations, covering the topic of differential equations. It appeared in the 2025 exam.

Looking for more practice? Explore all mathematics questions or browse differential equations questions on RankGuru.

Rank Guru

Differential Equations

Select the correct option:

Solution

🔒 Solution Hidden from View

More differential equations Practice Questions

Let $f (x)$ be differentiable s.t. $f^{'} (x) = 2 - f (x) / x$ and $f (1) = 1$ . Then $f (x)$ is:

The solution of the differential equation $d y / d x + y / x = x^{2}$ is:

About This Question

Solution

Differential Equations

Select the correct option:

Solution

🔒 Solution Hidden from View

More differential equations Practice Questions

Let f(x) be differentiable s.t. f′(x)=2−f(x)/x and f(1)=1. Then f(x) is:

The solution of the differential equation dy/dx+y/x=x2 is:

About This Question

Solution

Let $f (x)$ be differentiable s.t. $f^{'} (x) = 2 - f (x) / x$ and $f (1) = 1$ . Then $f (x)$ is:

The solution of the differential equation $d y / d x + y / x = x^{2}$ is: