Question

Solve the differential equation
x^2 dy/dx = 2y

Answer 1

Answer: The solution of the given differential equation is

$y=c e^{\frac{-2}{x} }$

Step-by-step explanation:

Explanation:-

Step:1

Given differential equation is

x^2 d y /d x = 2  y

By using variable separable that is separating x terms and y terms

$\frac{d y}{2 y} =\frac{d x}{x^2}$

Step 2:-

Now integrating on both sides we get the solution

$\int\ {\frac{1}{y} } \, d y =\int\limits {\frac{2 d x}{x^2} }$.........(1)

$By using integration formula$

$\int\limits{\frac{1}{y} } \, d y =logy+c$......(a)

and By using again integration formula

$\int\limits {x^n} \, d x = \frac{x^n}{n+1}$.....(b)

we will apply these (a)and (b) integration formulas in equation (1)

we get solution is

$logy = 2 \frac{x^{-2+1} }{-2+1} +log c$

(here you can take integration constant is log c)

Step 3:-

Now simplification

$logy - log c = \frac{-2}{x}$........(2)

Here you can apply logarithmic formula

$log a-log b=log(\frac{a}{b)}$

now the equation (2) can be written as

$log(\frac{y}{c}) = \frac{-2}{x}$......(3)

and again simplify using logarithmic formula

taking logarithmic function base (e) and equating b value

now $a = e^{b}$........(c)

by using equation (c) formula

now the equation (3) can be written as

$\frac{y}{c} = e^{\frac{-2}{x} }$

now $y= c e^{\frac{-2}{x} }$

Final answer :-

$y= c e^{\frac{-2}{x} }$