Матан

[fortpost]

10. 1+(y')²=2yy''

y'=p(y)
y''=pdp/dy

1+p²=2ypdp/dy
dy/y=2pdp/(1+p²)
∫dy/y=∫2pdp/(1+p²)
ln|y|=ln(1+p²)+lnC₁
ln|y|=ln|C₁(1+p²)|
y=C₁(1+p²)
y/C₁-1=p

y'=√(y/C₁-1)
dy/√(y/C₁-1)=dx
x=∫dy/√(y/C₁-1)=2C₁√(y/C₁-1)+C₂

[fortpost]

11. (y')²-yy''=xyy'/(x²+1)

y'/y-y''/y'=x/(x²+1)
ln|y|-ln|y'|=(1/2)ln(x²+1)+(1/2)lnC
(y/y')²=C(x²+1)
y/y'=C₁√(x²+1)
dy/y=dx/(C₁√(x²+1))
ln|y|=(1/C₁)ln(x+√(x²+1))+lnC₂
y=C₂(x+√(x²+1))^C₁₁

[fortpost]

12. y'''=x+sinx, y(0)=y'(0)=y''(0)=1

y''=∫(x+sinx)dx=x²/2-cosx+C₁
y'=∫(x²/2-cosx+C₁)dx=x³/6-sinx+C₁x+C₂
y=∫(x³/6-sinx+C₁x+C₂)dx=x⁴/24+cosx+C₁x²/2+C₂x+C₃

y''(0)=0-1+C₁=1 -> C₁=2
y'(0)=0-0+0+C₂=1 -> C₂=1
y(0)=0+1+0+0+C₃=1 -> C₃=0

y=x⁴/24+cosx+2x²/2+x=x⁴/24+cosx+x²+x

[fortpost]

13. y'y'''-3y''²=0, y(0)=0, y'(0)=1, y''(0)=1

y'=q(x)
qq''-3q'²=0

q'=p(q)
q''=pdp/dq

qpdp/dq-3p²=0
qdp/dq-3p=0
dp/p=3dq/q
ln|p|=3ln|q|+lnC₁
p=C₁q³
q'=C₁q³
dq/dx=C₁q³
dq/q³=C₁dx
C₁x=-1/(2q²)+C₂
q=1/√(2(C₂-C₁x))=1/√(C₂₁-C₁₁x)

y=∫dx/√(C₂₁-C₁₁x)=-(2/C₁₁)√(C₂₁-C₁₁x)+C₃

y'(0)=1/√C₂₁=1 -> C₂₁=1
y''(0)=(C₁₁/2)/√(C₂₁)³=C₁₁/2=1 -> C₁₁=2
y(0)=-(2/C₁₁)√(C₂₁)+C₃=(-1)1+C₃=0 -> C₃=1

y=-√(1-2x)+1

[fortpost]

14. y''+2y'+y=-e^-x+2xe^x

k²+2k+1=0
k_1,2=-1±√(1-1)=-1

y₀=C₁e^-x+C₂xe^-x
y=C₁(x)e^-x+C₂(x)xe^-x

C'₁(x)e^-x+C'₂(x)xe^-x=0
-C'₁(x)e^-x+C'₂(x)(e^-x-xe^-x)=-e^-x+2xe^x

C'₁(x)=x-2x²e^2x
C'₂(x)=-1+2xe^2x

C₁(x)=∫(x-2x²e^2x)dx=x²/2-x²e^2x+xe^2x-1/2e^2x+C₁₁
C₂(x)=∫(-1+2xe^2x)dx=xe^2x-1/2e^2x-x+C₂₁

y=(x²/2-x²e^2x+xe^2x-1/2e^2x+C₁₁)e^-x+(xe^2x-1/2e^2x-x+C₂₁)xe^-x

[fortpost]

15. y''+y=cosx+cos2x, y(0)=1, y'(0)=0

k²+1=0 -> k₁=i, k₂=-i

y₀=C₁cosx+C₂sinx
y=C₁(x)cosx+C₂(x)sinx

C'₁(x)cosx+C'₂(x)sinx=0
-C'₁(x)sinx+C'₂(x)cosx=cosx+cos2x
C'₁(x)=-sinxcosx-sinxcos2x
C'₂(x)=cos²x+cosxcos2x
C₁=∫(-sinxcosx-sinxcos2x)dx=1/2cos²x+1/6cos3x-1/2cosx+C₁₁
C₂=∫(cos²x+cosxcos2x)dx=x/2+1/4sin2x+1/2sinx+1/6sin3x+C₁₂

y=(1/2cos²x+1/6cos3x-1/2cosx+C₁₁)cosx+(x/2+1/4sin2x+1/2sinx+1/6sin3x+C₁₂)sinx
y'=-1/2sinxcos²x-1/3sin3xcosx+1/3sinxcos3x+sinxcosx-C₁₁sinx+1/2sinx+1/2xcosx-1/2sin³x+C₁₂cosx

y(0)=1/2+1/6-1/2+C₁₁=1 -> C₁₁=5/6
y'(0)=C₁₂=0 -> C₁₂=0

y=(1/2cos²x+1/6cos3x-1/2cosx+5/6)cosx+(x/2+1/4sin2x+1/2sinx+1/6sin3x)sinx

[fortpost]

16. y'''-3y''-y'+3y=2cosx+2

k³-3k²-k+3=0
k²(k-3)-(k-3)=0
(k²-1)(k-1)=0
k₁=-1, k₂=1, k₃=3

y₀=C₁e^-x+C₂e^x+C₃e^3x
y=C₁(x)e^-x+C₂(x)e^x+C₃(x)e^3x

C'₁(x)e^-x+C'₂(x)e^x+C'₃(x)e^3x=0
-C'₁(x)e^-x+C'₂(x)e^x+3C'₃(x)e^3x=0
C'₁(x)e^-x+C'₂(x)e^x+9C'₃(x)e^3x=2cosx+2

C'₁=1/4(cosx+1)e^x
C'₂=-1/2(cosx+1)e^-x
C'₃=1/4(cosx+1)e^-3x

C₁=∫1/4(cosx+1)e^xdx=1/4(1/2e^xcosx+1/2e^xsinx+e^x+C₁₁)
C₂=∫-1/2(cosx+1)e^-xdx=-1/2(-1/2e^-xcosx+1/2e^-xsinx-e^-x+C₂₁)
C₃=∫1/4(cosx+1)e^-3xdx=1/4(-3/10e^-3xcosx+1/10e^-3xsinx-1/3e^-3x+C₃₁)

y=3/10cosx-1/10sinx+2/3+1/4C₁₁e^-x-1/2C₂₁e^x+1/4C₃₁e^3x

[fortpost]

4.

a) y=x²sin((1+2x)/(1-2x))

dy=(x²sin((1+2x)/(1-2x)))'dx=(2x·sin((1+2x)/(1-2x))+x²·cos((1+2x)/(1-2x))·4/(1-2x)²)dx


б) y=√x·e^-tgx/arcsin²3x

dy=(√x·e^-tgx/arcsin²3x)'dx=((√x·e^-tgx)'·arcsin²3x-√x·e^-tgx·(arcsin²3x)')/arcsin⁴3x·dx=
((1/(2√x)·e^-tgx-√x·e^-tgx/cos²x)·arcsin²3x-√x·e^-tgx·6arcsin3x/√(1-9x²))/arcsin⁴3x·dx