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Exam 35: Differential Equations

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Question 51

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Use Euler's method to find an approximate solution to the differential equation $\frac{d y}{d x}=\frac{x^{2}}{y}$ from the boundary value $(1,1)$ to $x=5$ . Increase $x$ in steps of $\Delta x=1$ unit.

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Question 52

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Solve the differential equation: $y^{\prime \prime \prime}+y^{\prime}=2-\cos x$

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Question 53

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For an RLC circuit, $R=2.21 \Omega, C=105 \mu \mathrm{F}, L=0.100 \mathrm{H}$ , and $E=130 \mathrm{~V}$ . The current and charge are zero, when $t=0$ . Find $i$ as a function of time.

Solve: $y^{\prime \prime}=e^{3 x}$

In an RLC circuit, $R=25 \Omega, L=0.35 \mathrm{H}, C=2.0 \times 10^{-3} \mathrm{~F}$ , and $E=220 \sin 4 t$ . Find the amplitude of the steady-state current.

Solve: $y^{\prime \prime}-4 y=5$

Find the particular solution to: $y^{\prime \prime}-9 y=2 x+1, y=5$ and $y^{\prime}=3$ when $x=0$

Find the general solution: $y^{\prime \prime}=2 y^{\prime}+15 y$

Find the equation of the curve that passes through $(0,5)$ whose slope is $y^{\prime}=x-y$ .

The switch in the figure below is closed at $t=0$ . If $E=4 \mathrm{~V}, \mathrm{R}=5.0$ ohms, and $L=3.30$ henrys, find the current through the inductor at $t=1 \mathrm{~s}$ .

Solve the differential equation: $3 \frac{d y}{d x}=4 x-x^{3}$

Find the general solution: $y d x=4\left(x^{2}+y^{2}\right) d x+x d y$

Find the general solution: $\frac{2 d y}{d x}+\frac{4 y}{x}=6 \sin x$

Find the general solution: $5 y^{\prime \prime}+6 y^{\prime}+1 y=0$

Use Euler's method to graphically solve the differential equation $\frac{d y}{d x}=\frac{x^{2}}{y}$ from the boundary value $(1,1)$ to $x=5$ . Increase $x$ in steps of $\Delta x=1$ unit.

A metal ingot is taken out of a furnace when it reaches $950^{\circ} \mathrm{C}$ above room temperature. If it cools at $4 \% / minutes$ , find its temperature (above room temperature) 1 hour later. Round your answer to three significant digits.

Find the general solution: $2 y^{\prime \prime}+3 y^{\prime}-5 y=0$

Find the general solution of the differential equation: $y^{\prime}-3 x^{2} y=0$

Find the general solution to the first-order differential equation: $y-x y^{\prime}=y^{2} e^{x}$

Solve the differential equation: $2 y^{\prime \prime}-5 y^{\prime}-3 y=0$ , for $y^{\prime}(0,1)=2$