Love Riley
07/27/2023 · Junior High School
\( \operatorname { de } y = \frac { 5 x } { \sqrt { x - 2 } } \)
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Tutor-Verified Answer
Step-by-step Solution
To solve the differential equation
\[
\frac{dy}{dx} = \frac{5x}{\sqrt{x - 2}},
\]
we can integrate both sides with respect to \(x\).
1. **Separate the variables**: We can rewrite the equation as:
\[
dy = \frac{5x}{\sqrt{x - 2}} \, dx.
\]
2. **Integrate both sides**: We need to integrate the right-hand side:
\[
y = \int \frac{5x}{\sqrt{x - 2}} \, dx.
\]
To solve this integral, we can use a substitution. Let:
\[
u = x - 2 \quad \Rightarrow \quad du = dx \quad \Rightarrow \quad x = u + 2.
\]
Now, substituting \(x\) in terms of \(u\):
\[
y = \int \frac{5(u + 2)}{\sqrt{u}} \, du.
\]
This can be split into two integrals:
\[
y = \int \frac{5u}{\sqrt{u}} \, du + \int \frac{10}{\sqrt{u}} \, du.
\]
Simplifying the first integral:
\[
\int \frac{5u}{\sqrt{u}} \, du = \int 5u^{1/2} \, du = 5 \cdot \frac{u^{3/2}}{3/2} = \frac{10}{3} u^{3/2}.
\]
For the second integral:
\[
\int \frac{10}{\sqrt{u}} \, du = 10 \cdot 2\sqrt{u} = 20\sqrt{u}.
\]
Putting it all together, we have:
\[
y = \frac{10}{3} u^{3/2} + 20\sqrt{u} + C,
\]
where \(C\) is the constant of integration. Now, substituting back \(u = x - 2\):
\[
y = \frac{10}{3} (x - 2)^{3/2} + 20\sqrt{x - 2} + C.
\]
Thus, the solution to the differential equation is:
\[
y = \frac{10}{3} (x - 2)^{3/2} + 20\sqrt{x - 2} + C.
\]
Quick Answer
The solution to the differential equation is \( y = \frac{10}{3} (x - 2)^{3/2} + 20\sqrt{x - 2} + C \).
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