Explained: "Solve the following differential equation by the form..."

High School
Calculus and Analysis
Calculus 2
Differential equations

Pablo Dennis

Answered question

2022-03-20

Solve the following differential equation by the form of homogeneous equation. Letting

y = v x

The equation:

x^{2} \frac{d y}{d x} + x y + 1 = 0

Answer & Explanation

Jazlyn Mitchell

Beginner2022-03-21Added 14 answers

Step 1
Instead, let

v = x y

such that

\frac{d v}{d x} = y + x \frac{d y}{d x}

x^{2} \frac{d y}{d x} + x y = x \frac{d v}{d x}

and so

x^{2} \frac{d y}{d x} + x y + 1 = x \frac{d v}{d x} + 1 = 0

You can solve

x \frac{d v}{d x} = - 1

by separating.

glikozyd3s68

Beginner2022-03-22Added 16 answers

Step 1
Divide the proposed DE by x in order to obtain
$x^{2} y^{'} + x y + 1 = 0 ⟷ x y^{'} + y = - \frac{1}{x}$
$\leftrightarrow {(x y)}^{'} = - \frac{1}{x}$
$\leftrightarrow x y = - \ln | x | + k$
$\leftrightarrow y (x) = - \frac{\ln | x |}{x} + \frac{k}{x}$
and we are done.

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