Use the method for solving Bernoulli equations to solve the following differential equation.dy y+dx x-10=25(x-10)yIgnoring lost solutions, if any, the general solution is y =(Type an expression using x as the variable.)

Consider: dydx+yx-10=5x-10y12⇒1y12dydx+y12x-10=5x-10 ...(1) Let y12=t. So,…

Answered: Use the method for solving Bernoulli…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Use the method for solving Bernoulli equations to solve the following differential equation.
dy y
+
dx x-10
=
2
5(x-10)y
Ignoring lost solutions, if any, the general solution is y =
(Type an expression using x as the variable.)

Expert Solution

Step 1

Consider:

$\frac{d y}{d x} + \frac{y}{x - 10} = 5 (x - 10) y^{\frac{1}{2}} \Rightarrow \frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} + \frac{y^{\frac{1}{2}}}{(x - 10)} = 5 (x - 10) . . . (1)$

Let $y^{\frac{1}{2}} = t$ .

So,

$y^{\frac{1}{2}} = t \Rightarrow \frac{d}{d x} (y^{\frac{1}{2}}) = \frac{d t}{d x} \Rightarrow \frac{1}{2 y^{\frac{1}{2}}} \frac{d y}{d x} = \frac{d t}{d x} \Rightarrow \frac{1}{y^{\frac{1}{2}}} \frac{d y}{d x} = 2 \frac{d t}{d x}$