Solve the first order ordinary differential equation initial value problem for y(x):
x^2y'+xy=xe^x , y(1)=e+1
I have no idea even where to start.
Can anyone help please?
Divide through by x and the left hand side is 'exact', i.e. the evident 'result' of a product rule differentiation.
Just in case a picture helps...
... is the product rule. Straight continuous lines differentiate downwards (anti-differentiate up) with respect to x.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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Hopefully you at least recognize that this is a first order LINEAR differential equation; if it were seperable it would just be a matter of arranging the parts and gathering like terms. Remember that if we have a linear equation of the form , we can solve it using an integrating factor . We see we have y', multiplied by a function of x, and y multiplied by another function of x. Dividing through by x-squared:
Now we have a linear differntial equation; our next step is to get an "integrating factor" (check your notes):
Multiplying both sides by the integrating factor:
It is here that you need to convince yourself that , is just y times the integrating factor x, differentiated with respect to x (xy = y+xy'). Therefore we can write the left hand side as:
Integrating both sides:
Now we just need to solve for C to get a specific solution to this differential, with x=1, and y=e+1:
Therefore our final solution is
See if you can try this one using the exact same methods: