Solve the following differential equations. $y \frac{\mathrm{d}y}{\mathrm{d}x} = \sin x$ \(\frac{1

Question

Solve the following differential equations. $y \frac{\mathrm{d}y}{\mathrm{d}x} = \sin x$ $\frac{1}{x} \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x^2 + 1}$ $\frac{y^2}{x^3} \frac{\mathrm{d}y}{\mathrm{d}x} = \ln x$ $e^x \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x}{y}$ $v^2 \frac{\mathrm{d}v}{\mathrm{d}t} = (2 + t)^3$ $\tan x \frac{\mathrm{d}y}{\mathrm{d}x} = 2y^2 \sec^2 x$

Accepted Answer

(a) $\frac12y^2=-\cos x+C$. (b) $y=\frac12\ln(x^2+1)+C$, $x\ne0$. (c) $\frac13y^3=\frac{x^4}{4}\ln x-\frac{x^4}{16}+C$, $x>0$. (d) $\frac12y^2=-(x+1)\mathrm e^{-x}+C$, $y\ne0$. (e) $\frac13v^3=\frac14(t+2)^4+C$. (f) $y=0$, or $y=\frac1{C-2\ln|\tan x|}$ on admissible intervals.

Differential Equations - Solving Differential Equations by Separation of Variables

Differential Equations - Solving Differential Equations by Separation of Variables

Solution:

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