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Chapter 4: Problem 10

\(x^{2} y \frac{d y}{d x}=(x+1)(y+1)\)

Short Answer

Expert verified
Answer: The general solution is \(y = xCe^{\frac{-1}{x}} - 1\).

Step by step solution

01

Rewrite the differential equation with separation of variables

Separate the variables by dividing both sides by \(xy\) and \((x+1)(y+1)\): \(\frac{dy}{y+1} = \frac{x+1}{x^2} dx\)
02

Integrate both sides

Integrate both sides with respect to their respective variables: \(\int \frac{dy}{y+1} = \int \frac{x+1}{x^2} dx\)
03

Evaluate the left integral

The left side is a simple natural logarithm: \(\ln|y+1| = \int \frac{x+1}{x^2} dx + C_1\)
04

Evaluate the right integral

Using partial fraction decomposition, we can rewrite the right-side integrand as: \(\frac{x+1}{x^2} = \frac{A}{x} + \frac{B}{x^2}\) Solving for \(A\) and \(B\), we find: \(A = 1\) and \(B = 1\) Now, integrate the right side: \(\int \frac{x+1}{x^2} dx = \int \left(\frac{1}{x}+\frac{1}{x^2}\right) dx = \ln |x| - \frac{1}{x} + C_2\)
05

Combine the constants and solve for y

Combine the constants \(C_1\) and \(C_2\) into a new constant, \(C\): \(\ln|y+1| = \ln |x| - \frac{1}{x} + C\) Now, solve for \(y\) by exponentiating both sides: \(y+1 = xCe^{\frac{-1}{x}}\) Finally, subtracting 1 from both sides, we get the general solution: \(y = xCe^{\frac{-1}{x}} - 1\) The general solution for the given first-order, nonlinear ordinary differential equation is: \(y = xCe^{\frac{-1}{x}} - 1\)

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