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\(\frac{dy}{dx}=sin5x\)\(y=-\frac{1}{5}cos5x+C\)
\(\frac{dy}{dx}=(x+1)^2\)\(y=\frac{(x+1)^3}{3}+C\)
\(dx+e^3xdy=0\)\(y=\frac{1}{3}e^{-3x}+C\)
\(dy-(y-1)^2dx=0\)\(y=1-\frac{1}{x+C}\)
\(x\frac{dy}{dx}=4y\)\(y=Cx^4\)
\(\frac{dy}{dx}+2xy^2=0\)\(y=\frac{1}{x^2+C}\)
\(\frac{dy}{dx}=e^{3x+2y}\)\(-3e^{-2y}=2e^{3x}+C\)
\(e^xy\frac{dy}{dx}=e^{-y}+e^{-2x-2y}\)\(ye^y-e^y=-e^{-x}-\frac{1}{3}e^{-3x}+C\)
\(ylnx\frac{dx}{dy}=(\frac{y+1}{x})^2\)\(\frac{y^2}{2}+2y+ln|y|=\frac{x^3}{3}ln|x|-\frac{x^3}{9}+C\)
\(\frac{dy}{dx}=(\frac{2y+3}{4x+5})^2\)\(\frac{2}{2y+3}=\frac{1}{4x+5}+C\)
\(cscydx+sec^2xdy=0\)\(4cosy=2x+sin2x+C\)
\(sin3xdx+2ycos^33xdy\)\(y^2=-\frac{1}{6cos^23x}+C\)
\((e^y+1)^2e^{-y}dx+(e^x+1)^3e^{-x}dy=0\)\((e^x+1)^{-2}+2(e^y+1)^{-1}=C\)
\(x(1+y^2)^\frac{1}{2}dx=y(1+x^2)^\frac{1}{2}dy\)\(2\sqrt{1+y^2}=2\sqrt{1+x^2}+C\)
\(\frac{dS}{dr}=kS\)\(S=Ae^kr\)
\(\frac{dQ}{dt}=k(Q-70)\)\(Q=Ae^{kt}+70\)
\(\frac{dP}{dt}=P-P^2\)\(P=\frac{Ce^t}{1+Ce^t}\)
\(\frac{dN}{dt}+N=Nte^{t+2}\)\(N=Ce^{te^{t+2}-e^{t+2}-t}\)
\(\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}\)\((\frac{x+4}{y+3})^5=Ce^{x-y}\)
\(\frac{dy}{dx}=\frac{xy+2x-y-2}{xy-2x+4y-8}\)\(\frac{(y-1)^2}{(x-3)^5}=Ce^{x-y}\)
\(\frac{dy}{dx}=x\sqrt{1-y^2}\)\(y=sin({\frac{x^2}{2}+C})\)
\((e^x+e^{-x})\frac{dy}{dx}=y^2\)\(y=-\frac{1}{arctane^x+C}\)
\(\frac{dx}{dt}=4(x^2+1),\space x(\frac{\pi}{4})=1\)<div>\(x=tan(4t-\frac{3\pi}{4})\)<br></div>
\(\frac{dy}{dx}=\frac{y^2-1}{x^2-1}, \space y(2)=2\)\(y=x\)
\(x^2\frac{dy}{dx}=y-xy, \space y(-1)=-1\)\(y=\frac{1}{x}e^{-\frac{1}{x}-1}\)
\(\frac{dy}{dt}+2y=1,\space y(0)=\frac{5}{2}\)\(y=\frac{2}{e^{2t}}+\frac{1}{2}\)
\(\sqrt{1-y^2}dx-\sqrt{1-x^2}dy=0,\space y(0)=\frac{\sqrt{3}}{2}\)\(y=\frac{x}{2}+\frac{\sqrt{3}}{2}\sqrt{1-x^2}\)
\((1+x^4)dy+x(1+4y^2)dx=0,\space y(1)=0\)\(y=\frac{1}{2}\frac{1-x^2}{1+x^2}\)
\(\frac{dy}{dx}=5y\)\(y=Ce^{5x}\)
\(\frac{dy}{dx}+2y=0\)\(y=Ce^{-2x}\)
\(\frac{dy}{dx}+y=e^{3x}\)\(y=\frac{1}{4}e^{3x}+Ce^{-x}\)
\(3\frac{dy}{dx}+12y=4\)\(y=\frac{1}{3}+Ce^{-4x}\)
\(y'+3x^2y=x^2\)\(y=\frac{1}{3}+Ce^{-x^3}\)
\(y'+2xy=x^3\)\(y=\frac{1}{2}x^2-\frac{1}{2}+Ce^{-x^2}\)
<div>\(x^2y'+xy=1\)<br></div>\(y=\frac{lnx}{x}+\frac{C}{x}\)
\(y'=2y+x^2+5\)\(y=-\frac{1}{2}x^2-\frac{1}{2}x-\frac{11}{4}+Ce^{2x}\)
\(x\frac{dy}{dx}-y=x^2sinx\)\(y=-xcosx+Cx\)
\(x\frac{dy}{dx}+2y=3\)\(y=\frac{3}{2}+Cx^{-2}\)
\(x\frac{dy}{dx}+4y=x^3-x\)\(y=\frac{1}{7}x^3-\frac{1}{5}x+Cx^{-4}\)
\((1+x)\frac{dy}{dx}-xy=x+x^2\)\(y=-\frac{x^2+3x+3+Ce^x}{x+1}\)
\(x^2y'+x(x+2)y=e^x\)\(y=\frac{x^{-2}e^x}{2}+Cx^{-2}e^{-x}\)
\(xy'+(1+x)y=e^{-x}sin2x\)\(y=x^{-1}e^{-x}sin^2x+Cx^{-1}e^{-x}\)
\(ydx-4(x+y^6)dy=0\)\(x=2y^6+Cy^4\)
\(ydx=(ye^y-2x)dy\)\(x=e^y-\frac{2e^y}{y}+\frac{2e^y}{y^2}+\frac{C}{y^2}\)
\(cosx\frac{dy}{dx}+sinxy=1\)\(y=sinx+Ccosx\)
\(cos^2xsinx\frac{dy}{dx}+(cos^3x)y=1\)\(y=cosx+\frac{C}{sinx}\)
\((x+1)\frac{dy}{dx}+(x+2)y=2xe^{-x}\)\((x+1)e^xy=x^2+C\)
\((x+2)^2\frac{dy}{dx}=5-8y-4xy\)\(y=\frac{5}{3(x+2)}+\frac{C}{(x+2)^4}\)
\(\frac{dr}{d\theta}+rsec\theta=cos\theta\)\((sec\theta+tan\theta)r=\theta-cos\theta+C\)
\(\frac{dP}{dt}+2tP=P+4t-2\)\(P=2+Ce^{t-t^2}\)
\(x\frac{dy}{dx}+(3x+1)y=e^{-3x}\)\(y=e^{-3x}+Cx^{-1}e^{-3x}\)
\((x^2-1)\frac{dy}{dx}+2y=(x+1)^2\)\(y=\frac{x^2+x}{x-1}+\frac{C(x+1)}{x-1}\)
\(xy'+y=e^x,\space y(1)=2\)\(y=e^xx^{-1}+(2-e)x^{-1}\)
\(y\frac{dx}{dy}-x=2y^2,\space y(1)=5\)\(x=2y^2-\frac{49}{5}y\)
\(L\frac{di}{dt}+Ri=E,\space i(0)=i_0\)\(\frac{E}{R}+(i_0-\frac{E}{R})e^{-\frac{R}{L}t}\)
\(\frac{dT}{dt}=k(T-T_m),\space T(0)=T_0\)\(T=T_m+(T_0-T_m)Ce^{kt}\)
\((x+1)\frac{dy}{dx}+y=lnx,\space y(1)=10\)\(y=\frac{xlnx}{x+1}-\frac{x}{x+1}+\frac{21}{x+1}\)
\(y'+(tanx)y=cos^2x,\space y(0)=-1\)\(y=sinxcosx-cosx\)
\((\frac{e^{-2\sqrt{x}}-y}{\sqrt{x}})\frac{dx}{dy}=1,\space y(1)=1\)\(y=2\sqrt{x}e^{-2\sqrt{x}}+(e^2-2)e^{-2\sqrt{x}}\)
\((1+t^2)\frac{dx}{dt}+x=tan^{-1}t,\space x(0)=4\)\(x=(tan^{-1}-1)+5e^{-tan^{-1}t}\)
\((2x-1)dx+(3y+7)dy=0\)\(x^2-x+\frac{3}{2}y^2+7y=c\)
\((5x+4y)dx+(4x-8y^3)dy=0\)\(\frac{5}{2}x^2+4xy-2y^4=c\)
\((siny-ysinx)dx+(cosx+xcosy-y)dy=0\)\(xsiny+ycosx-\frac{1}{2}y^2=c\)
\((2xy^2-3)dx+(2x^2y+4)dy=0\)\(x^2y^2-3x+4y=c\)
\((1+lnx+\frac{y}{x})dx=(1-lnx)dy\)\(xlnx+ylnx-y=c\)
\((x-y^3+y^2sinx)dx=(3xy^2+2ycosx)dy\)\(\frac{x^2}{2}-xy^3-y^2cosx=c\)
\((x^3+y^3)dx+3xy^2dy=0\)\(\frac{x^4}{4}+xy^3=c\)
\((3x^2y+e^y)dx+(x^3+xe^y-2y)dy\)\(x^3y+e^yx-y^2=c\)
\(x\frac{dy}{dx}=2xe^x-y+6x^2\)\(xy-2xe^x+2e^x-2x^3=c\)
\((1-\frac{3}{y}+x)\frac{dy}{dx}+y=\frac{3}{x}-1\)\(3lnx-x-xy+3ln|y|-y=c\)
\((5y-2x)y'-2y=0\)\(2xy-\frac{5}{2}y^2=c\)
\((tanx-sinxsiny)dx+cosxcosydy=0\)\(-ln|cosx|+cosxsiny=c\)
\((2ysinxcosx-y+2y^2e^{xy^2})dx=(x-sin^2x-4xye^{xy^2})dy\)\(ysin^2x-xy+2e^{xy^2}=c\)
\((4t^3y-15t^2-y)dt+(t^4+3y^2-t)dy=0\)\(t^4y-5t^3-ty+y^3=c\)
\((x+y)^2dx+(2xy+x^2-1)dy=0,\space y(1)=1\)\(\frac{1}{3}x^3+x^2y+xy^2-y=\frac{4}{3}\)
\((e^x+y)dx+(2+x+ye^y)dy=0, \space y(0)=1\)\(e^x+xy+2y-e^y+ye^y=c\)
\((4y+2t-5)dt+(6y+4t-1)dy=0,\space y(-1)=2\)\(4ty+t^2-5t+3y^2-y=8\)
\((\frac{3y^2-t^2}{y^5})\frac{dy}{dt}+\frac{t}{2y^4}=0,\space y(1)=1\)<div>\(\frac{t^2}{4y^4}-\frac{3}{2y^2}=-\frac{5}{4}\)<br></div>
\((y^2cosx-3x^2y-2x)dx+(2ysinx-x^3+lny)dy=0,\space y(0)=e\)\(y^2sinx-x^3y-x^2+ylny-y=0\)
\((\frac{1}{1+y^2}+cosx-2xy)\frac{dy}{dx}=y(y+sinx),\space y(0)=1\)\(xy^2-ycosx-tan^{-1}y+1+\frac{\pi}{4}=0\)
\((-xysinx+2ycosx)dx+2xcosxdy=0\)\(x^2y^2cosx=C\)
\((x^2+2xy-y^2)dx+(y^2+2xy-x^2)dy=0\)\(x^2+y^2=c(x+y)\)
\((2y^2+3x)dx+2xydy=0\)\(x^2y^2+x^3=c\)marked
\(y(x+y+1)dx+(x+2y)dy=0\)\(xye^x+y^2e^x=c\)
\(6xydx+(4y+9x^2)dy=0\)\(y^4+3x^2y^3=c\)
\(cosxdx+(1+\frac{2}{y})sinxdy=0\)\(y^2e^ysinx=c\)
\((10-6y+e^{-3x})dx-2dy=0\)\(\frac{10}{3}e^{3x}-2ye^{3x}+x=c\)
\((y^2+xy^3)dx+(5y^2-xy+y^3siny)dy=0\)\(\frac{x}{y}+\frac{x^2}{2}+5ln|y|-cosy=c\)
\(xdx+(x^2y+4y)dy=0,\space y(4)=0\)\((x^2+4)e^{y^2}=20\)
\((x^2+y^2-5)dx=(y+xy)dy,\space y(0)=1\)\(-\frac{y^2}{2(1+x)^2}+ln|1+x|+\frac{2}{1+x}+\frac{2}{(1+x)^2}=\frac{7}{2}\)
\((x-y)dx+xdy=0\)\(y=-xln|x|+Cx\)
\((x+y)dx+xdy=0\)\(y=\frac{Cx^{-1}-x}{2}\)
\(xdx+(y-2x)dy=0\)\((y-x)ln|y-x|+x=C(y-x)\)
\(ydx=2(x+y)dy\)\(x+2y=cy^2\)
\((y^2+yx)dx-x^2dy=0\)\(yln|x|+x=cy\)
\((y^2+yx)dx+x^2dy=0\)\(x^2y=c(y+2x)\)
\(\frac{dy}{dx}=\frac{y-x}{y+x}\)\(ln|y^2+x^2|+2tan^{-1}(\frac{y}{x})=c\)
\(\frac{dy}{dx}=\frac{x+3y}{3x+y}\)\((y-x)^2=c(y+x)\)
\(-ydx+(x+\sqrt{xy})dy=0\)\(y(ln|y|-c)^2=4x\)
\(x\frac{dy}{dx}=y+\sqrt{x^2-y^2},\space x>0\)\(y=xsin(ln|x|+c)\)
\(xy^2\frac{dy}{dx}=y^3-x^3,\space y(1)=2\)\(3x^3ln|x|+y^3=8x^3\)
\((x^2+2y^2)\frac{dx}{dy}=xy,\space y(-1)=1\)\(2x^4=y^2+x^2\)
\(ydx+x(lnx-lny-1)dy=0,\space y(1)=e\)\(yln|\frac{y}{x}|=-e\)
\((x+ye^{\frac{y}{x}})dx-xe^{\frac{y}{x}}dy=0,\space y(1)=0\)\(ln|x|=e^{\frac{y}{x}}-1\)
\(x\frac{dy}{dx}+y=\frac{1}{y^2}\)\(y^3=1+cx^{-3}\)
\(\frac{dy}{dx}-y=e^xy^2\)\(y=\frac{1}{ce^{-x}-\frac{1}{2}e^x}\)
\(\frac{dy}{dx}=y(xy^3-1)\)\(y=\sqrt[3]{\frac{1}{x+\frac{1}{3}+ce^{3x}}}\)
\(x\frac{dy}{dx}-(1+x)y=xy^2\)\(y=\frac{xe^x}{e^x-xe^x+c}\)
\(t^2\frac{dy}{dt}+y^2=ty\)\(y=\frac{t}{ln(ct)}\)
\(3(1+t^2)\frac{dy}{dt}=2ty(y^3-1)\)\(y^3=\frac{1}{1+c(1+t^2)}\)
\(x^2\frac{dy}{dx}-2xy=3y^4,\space y(1)=\frac{1}{2}\)\(y^3=\frac{1}{\frac{49}{5}x^{-6}-\frac{9}{5}x^{-1}}\)
\(y^{\frac{1}{2}}\frac{dy}{dx}+y^{\frac{3}{2}}=1,\space y(0)=4\)\(y^{\frac{3}{2}}=1+7e^{-\frac{3}{2}x}\)
\(\frac{dy}{dx}=(x+y+1)^2\)\(y=tan(x+c)-x-1\)
\(\frac{dy}{dx}=\frac{1-x-y}{x+y}\)\((x+y)^2=2x+c\)
\(\frac{dy}{dx}=tan^2(x+y)\)\(2y+sin(2x+2y)=2x+c\)
\(\frac{dy}{dx}=sin(x+y)\)\(tan(x+y)-sec(x+y)=x+c\)
\(\frac{dy}{dx}=2+\sqrt{y-2x+3}\)\(2\sqrt{y-2x+3}=x+c\)
\(\frac{dy}{dx}=1+e^{y-x+5}\)\(-e^{x-y-5}=x+c\)
\(\frac{dy}{dx}=cos(x+y),\space y(0)=\frac{\pi}{4}\)\(csc(x+y)-cot(x+y)=x+\sqrt{2}-1\)
\(\frac{dy}{dx}=\frac{3x+2y}{3x+3y+2},\space y(-1)=-1\)\(5y-5x+2ln|75x+50y+30|=2ln95\)
Find \(y_2\) where \(y''-4y'+4y=0\) and \(y_1=e^{2x}\)\(y_2=xe^{2x}\)
Find \(y_2\) where \(y''+2y'+y=0\) and \(y_1=xe^{-x}\)\(y_2=e^{-x}\)
Find \(y_2\) where \(y''+16y=0\) and \(y_1=cos4x\)\(y_2=sin4x\)
Find \(y_2\) where \(y''+9y=0\) and \(y_1=sin3x\)\(y_2=cos3x\)
Find \(y_2\) where \(y''-y=0\) and \(y_1=coshx\)\(y_2=sinhx\)
Find \(y_2\) where \(y''-25y=0\) and \(y_1=e^{5x}\)\(y_2=e^{-5x}\)
Find \(y_2\) where \(9y''-12y'+4y=0\) and \(y_1=e^{\frac{2x}{3}}\)\(y_2=xe^\frac{2x}{3}\)
Find \(y_2\) where \(6y''+y'-y=0\) and \(y_1=e^{\frac{x}{3}}\)\(y_2=e^{-\frac{x}{2}}\)
Find \(y_2\) where \(x^2y''-7xy'+16y=0\) and \(y_1=x^4\)\(y_2=x^4lnx\)
Find \(y_2\) where \(x^2y''+2xy'-6y=0\) and \(y_1=x^2\)\(y_2=x^{-3}\)
Find \(y_2\) where \(xy''+y'=0\) and \(y_1=lnx\)\(y_2=1\)
Find \(y_2\) where \(4x^2y''+y=0\) and \(y_1=x^{\frac{1}{2}}lnx\)\(y_2=x^{\frac{1}{2}}\)
Find \(y_2\) where \(x^2y''-xy'+2y=0\) and \(y_1=xsin(lnx)\)\(y_2=xcos(lnx)\)
Find \(y_2\) where \(x^2y''-3xy'+5y=0\) and \(y_1=x^2cos(lnx)\)\(y_2=x^2sin(lnx)\)
Find \(y_2\) where \((1-2x-x^2)y''+2(1+x)y'-2y=0\) and \(y_1=x+1\)\(y_2=x^2+x+2\)
Find \(y_2\) where \((1-x^2)y''+2xy'=0\) and \(y_1=1\)\(y_2=x-\frac{x^3}{3}\)
\(4y''+y'=0\)\(y=c_1+c_2e^{-\frac{x}{4}}\)
\(y''-36y=0\)\(y=c_1e^{6x}+c_2e^{-6x}\)
\(y''-y'-6y=0\)\(y=c_1e^{3x}+c_2e^{-2x}\)
\(y''-3y'+2y=0\)\(y=c_1e^{x}+c_2e^{2x}\)
\(y''+8y'+16y=0\)\(y=c_1e^{-4x}+c_2xe^{-4x}\)
\(y''-10y'+25y=0\)\(y=c_1e^{5x}+c_2xe^{5x}\)
\(12y''-5y'-2y=0\)\(y=c_1e^{\frac{2x}{3}}+c_2e^{-\frac{x}{4}}\)
\(y''+4y'-y=0\)\(y=c_1e^{(-2+\sqrt{5})x}+c_2e^{(-2-\sqrt{5})x}\)
\(y''+9y=0\)\(y=c_1cos3x+c_2sin3x\)
\(3y''+y=0\)\(y=c_1cos\frac{x}{\sqrt{3}}+c_2sin\frac{x}{\sqrt{3}}\)
\(y''-4y'+5y=0\)\(y=e^{2x}(c_1cosx+c_2sinx)\)
\(2y''+2y'+y=0\)\(y=e^{-\frac{x}{2}}(c_1cos\frac{x}{2}+c_2sin\frac{x}{2})\)
\(3y''+2y'+y=0\)<div>\(y=e^{-\frac{x}{3}}(c_1cos\frac{\sqrt{2}x}{3}+c_2sin\frac{\sqrt{2}x}{3})\)<br></div>
\(2y''-3y'+4y=0\)\(y=e^\frac{3x}{4}(c_1cos\frac{\sqrt{23}x}{4}+c_2sin\frac{\sqrt{23}x}{4})\)
\(y'''-4y''-5y'=0\)\(y=c_1+c_2e^{5x}+c_3e^{-x}\)
\(y'''-y=0\)\(y=c_1e^x+e^{-\frac{x}{2}}(c_2cos\frac{\sqrt{3}}{2}x+c_3sin\frac{\sqrt{3}}{2}x)\)
\(y'''-5y''+3y'+9y\)\(y=c_1e^{-x}+c_2e^{3x}+c_3xe^{3x}\)
\(y'''+3y''-4y'-12y=0\)\(y=c_1e^{2x}+c_2e^{-2x}+c_3e^{-3x}\)
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