To solve the differential equation \((x^2+1)\frac{dy}{dx} + 2xy = \sqrt{x^2+4}\), we can use the method of integrating factors for a linear first-order differential equation.
The given differential equation can be written in the standard form:
\(\frac{dy}{dx} + P(x)y = Q(x)\)
where \(P(x) = \frac{2x}{x^2+1}\) and \(Q(x) = \frac{\sqrt{x^2+4}}{x^2+1}\).
The integrating factor \(\mu(x)\) is given by:
\(\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{2x}{x^2+1} \, dx}\)
Calculating the integral:
\(\int \frac{2x}{x^2+1} \, dx = \ln |x^2+1|\)
Therefore, the integrating factor is:
\(\mu(x) = e^{\ln |x^2+1|} = |x^2+1|\)
Since \(x^2 + 1\) is always positive, we have:
\(\mu(x) = x^2+1\)
Multiply the entire differential equation by the integrating factor:
\((x^2+1)\frac{dy}{dx} + 2x(x^2+1)y = \sqrt{x^2+4}\)
The left side of the equation becomes the derivative of \(((x^2+1)y)\):
\(\frac{d}{dx}((x^2+1)y) = \sqrt{x^2+4}\)
Integrate both sides with respect to \(x\):
\int \sqrt{x^2+4} \, dx, perform a trigonometric substitution (e.g., \(x = 2\tan(\theta)\)):
This simplifies to:
y = (x^2+1)^{-1}(\frac{1}{2}x\sqrt{x^2+4}+2\ln |x+\sqrt{x^2+4}|) + C
Thus, the correct answer is:
\(y=(x^2+1)^{-1}(\frac{1}{2}x\sqrt{x^2+4}+2log|(x+\sqrt{x^2+4})|) + c; \text{ where c is a constant}\)
Important Questions from Algebra (Notes)
- What is the remainder when 2023²⁰²⁴ + 2025²⁰²⁴ is divided by 2024?
In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If she/he attempts all 60 questions and secures 130 marks, the number of questions she/he attempts wrongly, are?
Match List-I with List-II
List-1 List-II (A) If $\begin{bmatrix}\lambda-1 & 0 \\ 0 & \lambda-1 \end{bmatrix} $, then $\lambda$ is (I) 0 (B) If A=$ \begin{bmatrix}1 & 2 \\2 & 4 \end{bmatrix} $, then $\Delta$ is (II) 1 (C) If A = $ \begin{bmatrix}1 & 0 \\0 & \frac{1}{2} \end{bmatrix} $, then $|A^{-1}|$ is (III) -2 (D) If $ \begin{bmatrix}a+1 & 1 \\1 & 2 \end{bmatrix} = \begin{bmatrix}-1 & 1 \\1 & 2 \end{bmatrix} $, then a is (IV) 2 Choose the correct answer from the options given below:
- If (x - 1) is a factor of $2x^2 - 5x + k = 0$, then the value of k is:
- If $x = (2+\sqrt{3})^{\frac{1}{3}} + (2+\sqrt{3})^{-\frac{1}{3}}$ and $x^3-3x + k = 0$, then the value of k is:
