Substitution of variables in differential expressions.

Substitution of variables in differential expressions.

Often in differential expressions, derivatives with respect to certain variables need to be expressed in terms of derivatives with respect to new variables.

Examples.

1. Transform the equation x4d2ydx2+2x3dydxy=0, assuming x=1t.

Solution.

Let's express the derivatives of y with respect to x in terms of derivatives of y with respect to t:

dydx=dydtdxdt=dydt1t2=t2dydt,

d2ydx2=ddx(dydx)=ddt(dydx)dxdt=2tdydtt2d2ydt21t2=2t3dydt+t4d2ydt2.

Let's substitute the found values of derivatives and the expression x=1t into the given equation.

1t4(2t3dydt+t4d2ydt2)+2t3(t2)dydty=2tdydt+d2ydt22tdydty=0.

Therefore, d2ydt2y=0.

Answer: d2ydt2y=0.

2. Transform the equation 3(d2ydx2)2dydxd3ydx3d2ydx2(dydx)2=0, taking y as the argument.

Solution.

Let's express the derivatives of y with respect to x in terms of derivatives of x with respect to y: dydx=1dxdy,

d2ydx2=ddx(1dxdy)=ddy(1dxdy)dydx=d2xdy2(dxdy)21dxdy=d2xdy2(dxdy)3,

d3ydx3=ddx(d2xdy2(dxdy)3)=ddy(d2xdy2(dxdy)3)dydx=

=d3xdy3(dxdy)33(dxdy)2d2xdy2d2xdy2(dxdy)61dxdy

Let's substitute the obtained expressions of derivatives into the given equation. We get

3(d2xdy2(dxdy)3)21dxdy(d3xdy3(dxdy)33(dxdy)2d2xdy2d2xdy2(dxdy)61dxdy) (d2xdy2(dxdy)3)(1dxdy)2=0

3(d2xdy2)2(dxdy)6+d3xdy3dxdy3(d2xdy2)2(dxdy)6+d2xdy2(dxdy)5=0

d3xdy3+d2xdy2(dxdy)5=0

d3xdy3+d2xdy2=0.

Thus, we have obtained the solution.

Answer: d3xdy3+d2xdy2=0.

3. Transform the equation (xyy)2=2xy(1+y2), by switching to polar coordinates.

Solution.

We have

x=rcosφ,y=rsinφ,

dx=cosφdrrsinφdφ,dy=sinφdr+rcosφdφ,

y=dydx=sinφdr+rcosφdφcosφdrrsinφdφ.

We substitute the expressions for x,,,y,,,y into the given equation. (rcosφsinφdr+rcosφdφcosφdrrsinφdφrsinφ)2= =2r2cosφsinφ(1+(sinφdr+rcosφdφcosφdrrsinφdφ)2)

(rsinφcosφdr+r2cos2φdφrsinφcosφdr+r2sin2φdφcosφdrrsinφdφ)2=

=2r2cosφsinφ(cosφdrrsinφdφ)2+(sinφdr+rcosφdφ)2(cosφdrrsinφdφ)2

(r2dφcosφdrrsinφdφ)2=2r2cosφsinφ(dr2+r2dφ2)(cosφdrrsinφdφ)2

r4dφ2=r2sin2φdr2+r4sin2φdφ2

sin2φdr2=(1sin2φ)r2dφ2(drdφ)2=1sin2φsin2φr2

r2=1sin2φsin2φr2

Answer: r2=1sin2φsin2φr2

4. To transform the equation (x+y)zx(xy)zy=0, перейдя к новым независимым переменным u и v, если u=lnx2+y2,v=arctgyx.

using the new independent variables u and v, where u=lnx2+y2,,,v=arctanyx.

Solution.

We express the partial derivatives of z with respect to x and y in terms of the partial derivatives of z with respect to u and v.

We have ux=1x2+y2(x2+y2)x=1x2+y22x2x2+y2=xx2+y2;

uy=1x2+y2(x2+y2)y=1x2+y22y2x2+y2=yx2+y2;

vx=11+(yx)2(yx)x=11+(yx)2yx2=yx2+y2;

vy=11+(yx)2(yx)y=11+(yx)21x=xx2+y2.

Next, we find

zx=zuux+zvvx=zuxx2+y2+zvyx2+y2;

zy=zuuy+zvvy=zuyx2+y2+zvxx2+y2.

Let's substitute the found expressions of derivatives into the given equation:

(x+y)(zuxx2+y2+zvyx2+y2)(xy)(zuyx2+y2+zvxx2+y2)=0

zux2+xyxy+y2x2+y2+zvxyy2x2+xyx2+y2=0

zuzv=0.

Answer: zu=zv.

5. To transform the equation (xy+z)zx+(1y2)zy=x+yz, taking u=yzx,,,v=xzy as new independent variables and w=xyz as the new function.

Solution.

We express the partial derivatives zx and zy in terms of wu and wv.

We have:

du=dx+zdy+ydz;

dv=zdx+xdzdy;

dw=ydx+xdydz.

Taking into account the formula

dudt=ux1dx1dt+ux2dx2dt++uxndxndt, we find

dw=wudu+wvdv

ydx+xdydz=wu(dx+zdy+ydz)+wv(zdx+xdzdy)

ydx+xdywu(dx+zdy)wv(zdxdy)=dz(1+ywu+xwv)

dz=y+wuzwv1+ywu+xwvdx+xzwu+wv1+ywu+xwvdy

dzdx=y+wuzwv1+ywu+xwv;

dzdy=xzwu+wv1+ywu+xwv.

Let's substitute the found expressions for zx and zy into the given equation. We get

(xy+z)y+wuzwv1+ywu+xwv+(1y2)xzwu+wv1+ywu+xwv=x+yz

xy2+zy+xywu+zwuxyzwvz2wv+xzwu+wvxy2+y2zwuy2wv1+ywu+xwv= =x+yz

zy+xywuxyzwvz2wv+x+wv+y2zwuy2wv1+ywu+xwv=x+yz

zy+xywuxyzwvz2wv+x+wv+y2zwuy2wv= =(x+yz)(1+ywu+xwv)

zy+xywuxyzwvz2wv+x+wv+y2zwuy2wv=

=x+xywu+x2wv+yz+y2zwu+xyzwv

xyzwvz2wv+wvy2wv=x2wv+xyzwv.

Thus,

wv(x2+y2+z2+2xyz1)=0wv=0.

Answer: wv=0.

Tags: Substitution of variables, calculus, derivative, differential expressions, mathematical analysis