Method of variation of parameters
Use the method of variation of parameters to determine a general solution to differential equation: ty''+(1-2t)y'+(t-1)y=te^t given that y1=e^t and y2=e^t (ln t) are two linearly independent solutions to the corresponding homogenous equation.
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Solve the given differential equation by using the substitution u=y' (Use C1 and C2as arbitrary constants. Use x as the independent variable.)
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Suggest: f : R----> R is continuous, a, b are functions from R to R and are differentiable on R. g(x) = integral of f(t)dt with upper limit = b(x) and lower limit = a(x) the problem is to determine a general formula for the derivative of g at x
Prove the validity of the limit lim x->x_0 (ax+b)=ax_0+b by converting to a statement about sequences.
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The radioactive isotope of lead, Pb-209, decays at the rate proportional to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for 95% of the lead to decay?
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