What is a solution to the differential equation $y\prime =2x+1$?

blackdivcp
2022-10-11
Answered

What is a solution to the differential equation $y\prime =2x+1$?

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plomet6a

Answered 2022-10-12
Author has **20** answers

$\frac{dy}{dx}=2x+1$

$\int dy=\int 2x+1dx$

$y={x}^{2}+x+C$

$\int dy=\int 2x+1dx$

$y={x}^{2}+x+C$

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What is the solution to the Differential Equation $\frac{4}{{y}^{3}}\frac{dy}{dx}=\frac{1}{x}$?

asked 2022-06-22

I have a linear first order ordinary differential equation

$\frac{dy}{dx}+\mathrm{tan}(x)y=2{\mathrm{cos}}^{2}x\mathrm{sin}x-\mathrm{sec}x$

with an initial condition as $y(\frac{\pi}{4})=3\sqrt{2}$

My integrating factor $\mu (x)=\mathrm{sec}x$

After multiplication with the integration factor what I get is:

$(secx\text{}y{)}^{\prime}=sin2x-se{c}^{2}x$

or

$(secx\text{}y{)}^{\prime}=2sinxcosx-se{c}^{2}x$

If I use the first equation I get:

$y(x)=\frac{\frac{1}{2}cos2x-tanx+c}{secx}$

and using the second equation I get:

$y(x)=\frac{si{n}^{2}x-tanx+c}{secx}$

$\int 2\text{}sinx\text{}cosx\text{}dx=2\frac{si{n}^{2}x}{2}=si{n}^{2}x$

with the first equation I get c=7 and second equation I get $c=\frac{13}{2}$.

It is a very simple differential equation but when I solve it I get two different answers. Is this ok?

$\frac{dy}{dx}+\mathrm{tan}(x)y=2{\mathrm{cos}}^{2}x\mathrm{sin}x-\mathrm{sec}x$

with an initial condition as $y(\frac{\pi}{4})=3\sqrt{2}$

My integrating factor $\mu (x)=\mathrm{sec}x$

After multiplication with the integration factor what I get is:

$(secx\text{}y{)}^{\prime}=sin2x-se{c}^{2}x$

or

$(secx\text{}y{)}^{\prime}=2sinxcosx-se{c}^{2}x$

If I use the first equation I get:

$y(x)=\frac{\frac{1}{2}cos2x-tanx+c}{secx}$

and using the second equation I get:

$y(x)=\frac{si{n}^{2}x-tanx+c}{secx}$

$\int 2\text{}sinx\text{}cosx\text{}dx=2\frac{si{n}^{2}x}{2}=si{n}^{2}x$

with the first equation I get c=7 and second equation I get $c=\frac{13}{2}$.

It is a very simple differential equation but when I solve it I get two different answers. Is this ok?

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