# ODE collection of first order differential equations

ODE

Solution

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${y}^{\prime }\left(x\right)+ay\left(x\right)=b$

$y\left(x\right)=\frac{b}{a}+C{e}^{-ax}$

${y}^{\prime }\left(x\right)+ay\left(x\right)=c{e}^{bx}$

$y\left(x\right)=\frac{c{e}^{bx}}{a+b}+{k}_{1}{e}^{-ax}$

${y}^{\prime }\left(x\right)+2xy\left(x\right)=x{e}^{-{x}^{2}}$

$y\left(x\right)={k}_{1}{e}^{-{x}^{2}}+\frac{1}{2}{e}^{-{x}^{2}}{x}^{2}={e}^{-{x}^{2}}\left({k}_{1}+\frac{{x}^{2}}{2}\right)$

${y}^{\prime }\left(x\right)+xy\left(x\right)=x$

$y\left(x\right)={c}_{1}{e}^{-\frac{{x}^{2}}{2}}+1$

${y}^{\prime }\left(x\right)+y\left(x\right)=x$

$y\left(x\right)={c}_{1}{e}^{-x}+x-1$

${y}^{\prime }\left(x\right)={y\left(x\right)}^{2}$

$y\left(x\right)=\frac{1}{{c}_{1}-x}$

${y}^{\prime }\left(x\right)+{y\left(x\right)}^{2}=1$

Separated:$\frac{{y}^{\prime }\left(x\right)}{1-{y\left(x\right)}^{2}}=1$

Riccatische Equation:${y}^{\prime }\left(x\right)=-{y\left(x\right)}^{2}+1$

$y\left(x\right)=\frac{{e}^{2x}-{e}^{2{c}_{1}}}{{e}^{2{c}_{1}}+{e}^{2x}}$

${y}^{\prime }\left(x\right)=\left(Ay\left(x\right)-a\right)\left(By\left(x\right)-b\right)$

$y\left(x\right)=\frac{b{e}^{x\left(aB-Ab\right)}-aC}{B{e}^{x\left(aB-Ab\right)}-AC}$

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