Hi, I don't understand how I am meant to solve this:
$\displaystyle x\frac{dy}{dx} + y = x^4, y(1) = 1$
Any help appreciated
Hello, Beard!
$\displaystyle x\frac{dy}{dx} + y \:=\: x^4,\quad y(1) = 1$
Divide by $\displaystyle x\!:\quad \frac{dy}{dx} + \frac{1}{x}\,y \:=\:x^3$
Integrating Factor: .$\displaystyle I \:=\:e^{\int\frac{dx}{x}} \:=\:e^{\ln x} \:=\:x$
Multiply by $\displaystyle x\!:\quad x\,\frac{dy}{dx} + y \:=\:x^4$
We have: .$\displaystyle \frac{d}{dx}(xy) \:=\:x^4$
Integrate: .$\displaystyle xy \:=\:\frac{1}{5}x^5 + C \quad\Rightarrow\quad y \:=\:\frac{1}{5}x^4 + \frac{C}{x}$
From $\displaystyle y(1) = 1\!:\quad 1 \:=\:\tfrac{1}{5}(1^4) + \frac{C}{1} \quad\Rightarrow\quad C \:=\:\frac{4}{5}$
Therefore: .$\displaystyle y \;=\;\frac{1}{5}x^4 + \frac{4}{5x}$
It makes the left-hand side look like the result of a product-rule differentiation. Just in case a picture helps... suppose
is the product rule - straight continuous lines differentiating downwards (integrating up) with respect to x. Then we're trying to fit the left-hand side...
$\displaystyle \frac{dy}{dx} + y\ \frac{1}{x}$
... along the bottom row. As it comes, it won't fit (and satisfy the rule). But if we accomodate the chain rule...
... inside the (legs-uncrossed version of the) product rule...
... we can see how to fix it...
Spoiler:
Hope that helps.
Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
__________________________________________
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Balloon Calculus: standard integrals, derivatives and methods
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Fair enough! Try to focus on the product rule, one way or another... or, maybe not - you could simply practice applying the formula. Whatever works for you...
PS: are you clear, at least, that the equation is 'multiplied through' by the I.F.?
i.e. given...
A + B = C (1)
... it follows that
A * I.F. + B * I.F. = C * I.F. (2)
... and (2) is easier than (1)