| utf24-001 Solve the following differential equations subject to the prescribed initial conditions. \begin{alignat*}{6} &a) \frac{d y}{dx} = 4(x-7)^3 & &x=8, & &y=10 & &b) \frac{d y}{dx} =\frac{ x^2 + 1 }{ x^2} & &x=1, & &y=1 \\ &c) \frac{d y}{dx} = x\sqrt y & &x=0, & &y=1 & &b) \frac{d y}{dx} =\frac{ 4\sqrt{(1+y^2)^3} }{ y} & &x=0, & &y=1 \end{alignat*} |
| utf24-002 Find all continuous functions $ f( x)$ satisfying $$\int_{0}^{x}{ f( t)} dt = [ f( x)]^2 + C$$ |
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| utf26-004
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| utf28-001 Derive the formulas below. {(Hint: differentiate $g(g^{{-1}}(x))=x$ then use triangles (e.g. you know that $\arcsin x$ is an angle, $\theta$, whose sine is $x$.).)} $$a) \frac{ds}{dx}{\arccos x}=\frac{-1 }{ \sqrt {1-x^2}} b) \frac{ds}{dx}{\arctan x}= \frac{1 }{ 1+x^2} $$ |