Differential Equations

[Guest]

could someone explain if i have done this work example right as it would help me understand abit better?

x dy/dx+x^2=5

xy+x^2=5

y=5-x^2/x

y=5-x^2/x dx

∫(5-x^2)/x dx=5 ln⁡(x)-x^2/2+c_1

y=5 ln⁡(x)-x^2/2+c_1

[Guest]

could someone explain if i have done this work example right as it would help me understand abit better?

x dy/dx+x^2=5

xy+x^2=5

How did "dy/dx" become "y"?

y=5-x^2/x

Again "y" should be "dy/dx".
And "5- x^2/x" should be "(5- x^2)/x". The parentheses are essential!

y=5-x^2/x dx

"dx" just kind of magically appears here!
You should have "dy= (5- x^2)/x dx"

∫(5-x^2)/x dx=5 ln⁡(x)-x^2/2+c_1

It would be better to first write "∫(5/x- x)dx" so that 5ln(x)- x^2/2+ c_1 makes more sense!

y=5 ln⁡(x)-x^2/2+c_1

Yes, that is the general solution to the equation.