Question:
If the solution of the differential equation \(\frac{dy}{dx} = \frac{\alpha x^2 + 4x - 4}{2y - 4}\), when \(y(1) = 3\), is \(y^2 - 4y = x^3 + 2x^2 - 4x + c\), then the value of \(\alpha\) is equal to
If the solution of the differential equation \(\frac{dy}{dx} = \frac{\alpha x^2 + 4x - 4}{2y - 4}\), when \(y(1) = 3\), is \(y^2 - 4y = x^3 + 2x^2 - 4x + c\), then the value of \(\alpha\) is equal to
Show Hint
In questions where the solution form is given, you don't always need to solve for the constant \(c\) unless specifically asked. Focus on equating the coefficients of corresponding powers of \(x\) or \(y\).
Updated On: Jun 24, 2026
- 1
- -1
- -3
- 3
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
We solve the given first-order differential equation by separating the variables \(y\) and \(x\).
Then we integrate both sides and compare the resulting general solution with the one provided in the question to find the constant \(\alpha\).
Step 2: Key Formula or Approach:
1. Variable Separation: \(g(y) dy = f(x) dx\).
2. Integration: \(\int g(y) dy = \int f(x) dx\).
Step 3: Detailed Explanation:
The given differential equation is:
\[ \frac{dy}{dx} = \frac{\alpha x^2 + 4x - 4}{2y - 4} \]
Separating the variables:
\[ (2y - 4) dy = (\alpha x^2 + 4x - 4) dx \]
Integrating both sides:
\[ \int (2y - 4) dy = \int (\alpha x^2 + 4x - 4) dx \]
\[ y^2 - 4y = \frac{\alpha x^3}{3} + \frac{4x^2}{2} - 4x + c \]
\[ y^2 - 4y = \frac{\alpha}{3} x^3 + 2x^2 - 4x + c \]
Comparing this general solution with the given solution \(y^2 - 4y = x^3 + 2x^2 - 4x + c\):
The coefficient of \(x^3\) must match:
\[ \frac{\alpha}{3} = 1 \]
\[ \alpha = 3 \]
Step 4: Final Answer:
The value of \(\alpha\) is 3.
We solve the given first-order differential equation by separating the variables \(y\) and \(x\).
Then we integrate both sides and compare the resulting general solution with the one provided in the question to find the constant \(\alpha\).
Step 2: Key Formula or Approach:
1. Variable Separation: \(g(y) dy = f(x) dx\).
2. Integration: \(\int g(y) dy = \int f(x) dx\).
Step 3: Detailed Explanation:
The given differential equation is:
\[ \frac{dy}{dx} = \frac{\alpha x^2 + 4x - 4}{2y - 4} \]
Separating the variables:
\[ (2y - 4) dy = (\alpha x^2 + 4x - 4) dx \]
Integrating both sides:
\[ \int (2y - 4) dy = \int (\alpha x^2 + 4x - 4) dx \]
\[ y^2 - 4y = \frac{\alpha x^3}{3} + \frac{4x^2}{2} - 4x + c \]
\[ y^2 - 4y = \frac{\alpha}{3} x^3 + 2x^2 - 4x + c \]
Comparing this general solution with the given solution \(y^2 - 4y = x^3 + 2x^2 - 4x + c\):
The coefficient of \(x^3\) must match:
\[ \frac{\alpha}{3} = 1 \]
\[ \alpha = 3 \]
Step 4: Final Answer:
The value of \(\alpha\) is 3.
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Differential equations Questions
- An integrating factor of the differential equation $xdy - ydx + x^2e^xdx = 0$ is
- If $xy\, = \,A \,sinx \,+ \,B \,cos \,x$ is the solution of the differential equation $x\frac{d^{2}y}{dx^{2}}-5a\frac{dy}{dx}+xy=0$ then the value of $a$ is equal to
- The order of the differential equation $\left(\frac{d^{3}\, y }{dx^{3}}\right)^{2} + \left(\frac{d^{2}\,y}{dx}\right)^{2} + \left(\frac{dy}{dx}\right)^{5} = 0 $ is
- The solution of $\frac{dy}{dx} + y \, \tan \, x = \sec \, x, y (0) = 0$ is
- If $ {{c}_{1}},{{c}_{2}},{{c}_{3}},{{c}_{4}},{{c}_{5}} $ and $ {{c}_{6}} $ are constants, then the order of the differential equation whose general solution is given by $ y={{c}_{1}}cos $ $ (x+{{c}_{2}})+{{c}_{3}}\sin (x+{{c}_{4}})+{{c}_{5}}{{e}^{x}}+{{c}_{6}} $
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions