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Chapter 6: Problem 38

Solve the homogeneous differential equation. \(y^{\prime}=\frac{x^{2}+y^{2}}{2 x y}\)

Short Answer

Expert verified
The solutions to the homogeneous differential equation are: \(y = x\sqrt{e^{-x+C} - 1}\) and \(y = -x\sqrt{e^{-x+C} - 1}\).

Step by step solution

01

Start the Substitution

The first step is to substitute \(v = \frac{y}{x}\) which leads to \(y = vx\) into the given differential equation. Then replace \(y'\) by \(v+xv'\).
02

Substitute the Values into the Original Equation

Substitute these values into the original differential equation. The equation becomes \(v + xv' = \frac{x^2 + v^2x^2}{2vx}\). After multiplying by \(2vx\), the equation is then simplified into \(2v^2x^2 + 2x^2vv' = x^2 + v^2x^2\).
03

Simplify and Solve

Subtract \(x^2\) and \(v^2x^2\) from both sides and then divide by \(v^2x^2\), we can reduce this to a separable differential equation of the form \(2vv' = -1 - v^2\). Then separate variables and integrate: \(\int \frac{2v}{1 + v^2} dv = - \int dx\) . This will yield \( \ln |1 + v^2| = -x + C\), which simplifies to \(1 + v^2 = e^{-x+C}\). Replace v back into the equation to obtain a formula for y, i.e., \(1 + (\frac{y}{x})^2 = e^{-x+C}\). Solve for y.
04

Series Sqrt

Solve the equation for y, square root both sides to isolate y. Thus the solutions would be \(y = x\sqrt{e^{-x+C} - 1}\) and \(y = -x\sqrt{e^{-x+C} - 1}\).

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=f(x)\) is a solution of a first-order differential equation, then \(y=f(x)+C\) is also a solution.

Find the particular solution of the differential equation that satisfies the boundary condition. $$ \begin{array}{ll} \underline{\text { Function }} & \underline{\text { Differential Equation }} \\\ 2 x y^{\prime}-y=x^{3}-x &\quad y(4)=2 \end{array} $$

At time \(t=0\), a bacterial culture weighs 1 gram. Two hours later, the culture weighs 2 grams. The maximum weight of the culture is 10 grams. (a) Write a logistic equation that models the weight of the bacterial culture. (b) Find the culture's weight after 5 hours. (c) When will the culture's weight reach 8 grams? (d) Write a logistic differential equation that models the growth rate of the culture's weight. Then repeat part (b) using Euler's Method with a step size of \(h=1\). Compare the approximation with the exact answers. (e) At what time is the culture's weight increasing most rapidly? Explain.

Solve the first-order differential equation by any appropriate method. $$ \left(y^{2}+x y\right) d x-x^{2} d y=0 $$

Solve the first-order differential equation by any appropriate method. $$ (x+y) d x-x d y=0 $$
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