If $y(x)$ is the solution of the differential equation $x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2,$ and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is ____.
Correct Answer: 8
Approach Solution - 1
Solution of the Given Differential Equation
Given: The equation is:
\(x \, dy - (y^2 - 4y) \, dx = 0, \, x > 0\)
Step 1: Rewriting the equation
We start by rewriting the equation as:
\(\int \frac{dy}{y^2 - 4y} = \int \frac{dx}{x}\)
Step 2: Solving the left-hand side integral
Now we solve the left-hand side integral:
\(\int \frac{1}{y^2 - 4y} \, dy = 4 \int \frac{dx}{x}\)
We can factor the denominator and use partial fraction decomposition to solve the integral:
\(\log|y - 4| - \log|y| = 4 \log x + \log c\)
Step 3: Simplifying the equation
We can simplify the equation as follows:
\(\frac{|y - 4|}{|y|} = |y| x^4 \quad \text{(let} \, c = 1\text{)}\)
Step 4: Analyzing the equation for possible solutions
We now solve for \( y \) in terms of \( x \) and analyze the possible cases.
Case 1: If \( C = 1 \), we get:
\(y = \frac{4}{1 - x^4}\)
Substituting \( y(1) \) gives us a rejected solution since it leads to an undefined value.
Case 2: If \( C = 2 \), we get:
\(y = \frac{4}{1 + x^4}\)
Step 5: Calculating specific values
We can now calculate specific values for \( y \):
\(y(1) = 2\)
Next, we compute \( y(\sqrt{2}) \):
\(y(\sqrt{2}) = \frac{4}{1 + (\sqrt{2})^4} = \frac{4}{1 + 4} = \frac{4}{5}\)
Now, calculate the final answer:
\(10y(\sqrt{2}) = 10 \times \frac{4}{5} = 8\)
Conclusion
The final value of \( 10y(\sqrt{2}) \) is 8, hence the answer is \( \boxed{8} \).
Approach Solution -2
Given,
$x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2,$
\(\frac{dy}{y^2-4y}=\frac{dx}{x}\)
\(\frac{1}{4}(\frac{1}{y-4}-\frac{1}{y})dy=\frac{dx}{x}\)
Integrate both sides
\(\log_c|y-4|-\log_c|y|=4\log_ex+\log_ec\)
\(|\frac{y-4}{y}|=x^4\)
\(|y-4|=|y|x^4\)
\(y-4=yx^4\)
\(y-yx^4=4\)
\(y=\frac{4}{1-x^4}\)
y(1)=2
\(y(\sqrt2)=\frac{4}{5}\)
\(10y(\sqrt2)=8\)
So, the answer is 8.
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations