Answer to Question #255269 in Differential Equations for Anne

Question #255269

(Variable Separable). Determine the solution of the differential equation.

a.) α\alpha dβ\beta + β\betadα\alpha + α\alphaβ\beta(3dα\alpha + dβ\beta) =0


b.) (xy + x)dx = (x2y2 + x2 + y2 +1)dy


c.) dx = t(1 + t2) sec2x dt


1
Expert's answer
2021-10-25T14:47:00-0400

a)


α(1+β)dβ=β(1+3α)dα\alpha(1+\beta)d\beta=-\beta(1+3\alpha)d\alpha

1+ββdβ=1+3ααdα\dfrac{1+\beta}{\beta}d\beta=-\dfrac{1+3\alpha}{\alpha}d\alpha

Integrate


1+ββdβ=1+3ααdα\int \dfrac{1+\beta}{\beta}d\beta=-\int\dfrac{1+3\alpha}{\alpha}d\alpha

ln(β)+β=ln(α)3α+lnC\ln(|\beta|)+\beta=-\ln(|\alpha|)-3\alpha+\ln C

βeβ=Cαe3α\beta e^{\beta}=\dfrac{C}{\alpha e^{3\alpha}}

αβe3α+β=C\alpha \beta e^{3\alpha+\beta}=C

b)


(y2+1)(x2+1)dy=x(y+1)dx(y^2+1)(x^2+1)dy=x(y+1)dx

y2+1y+1dy=xx2+1dx\dfrac{y^2+1}{y+1}dy=\dfrac{x}{x^2+1}dx

Integrate


y2+1y+1dy=xx2+1dx\int\dfrac{y^2+1}{y+1}dy=\int\dfrac{x}{x^2+1}dx

y2+1y+1dy=y2+2y+12y2+2y+1dy\int\dfrac{y^2+1}{y+1}dy=\int\dfrac{y^2+2y+1-2y-2+2}{y+1}dy

=(y+1)dy2dy+2dyy+1=\int(y+1)dy-2\int dy+2\int \dfrac{dy}{y+1}

=y22+y2y+2ln(y)+C1=\dfrac{y^2}{2}+y-2y+2\ln(|y|)+C_1

y22y+2ln(y)=12ln(x2+1)+lnC\dfrac{y^2}{2}-y+2\ln(|y|)=\dfrac{1}{2}\ln(x^2+1)+\ln C

c)


cos2xdx=t(1+t2)dt\cos^2x dx = t(1 + t^2) dt

Integrate


cos2xdx=t(1+t2)dt\int\cos^2x dx =\int t(1 + t^2) dt

12(1+cos(2x))dx=(t+t3)dt\dfrac{1}{2}\int(1+\cos(2x))dx=\int(t+t^3)dt

12x+14sin(2x)=t22+t44+C\dfrac{1}{2}x+\dfrac{1}{4}\sin(2x)=\dfrac{t^2}{2}+\dfrac{t^4}{4}+C


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