2 diff. ligninger og 2 integraler.

[noooa]

diff 1) xy'-5y=2x^5
diff 2) x dy/dx=cos^2x

Integral 1) sin^7x cosx dx
Integral 2) te^3t dt

[Janhaa]

a)
diff 1) xy'-5y=2x^5

b)
diff 2) x dy/dx=cos^2x

c)
Integral 1) sin^7x cosx dx
d)
Integral 2) te^3t dt

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a)

[tex]xy `-5y=2x^5\;(*)[/tex]

Innfører:
y = uv og y ` = u' v + uv'

Setter dette inn i (*)

x(u' v + uv') - 5(uv) = 2x[sup]5[/sup]


I: x(uv ') - 5(uv) = 0

[tex]x{dv\over dx}=5v[/tex]

[tex]\int {dv\over v}[/tex][tex]\;=\;[/tex][tex]5\int {dx\over x}[/tex]

[tex]ln(v)=5\cdot ln(x)[/tex]

[tex]v\;=\;x^5[/tex]


II: x(u`v) = 2x[sup]5[/sup]

setter inn v = x[sup]5[/sup]

som gir ved integrering:

u = 2ln(x) + C

y = uv = [2ln(x) + C]x[sup]5[/sup]

[Janhaa]

diff 1) xy'-5y=2x^5
diff 2) x dy/dx=cos^2x
Integral 1) sin^7x cosx dx
Integral 2) te^3t dt

---------------------------------------------------------------

1)

[tex]I\;=\;\int sin^7x\cdot cos(x) dx[/tex]

[tex]u=sinx[/tex]

[tex]du=cos(x)dx[/tex]

[tex]I\;=\;\int u^7 du[/tex][tex]\;=\;[/tex][tex]{u^8\over 8}\;+\;C[/tex]

[tex]I\;=\;{sin^8x\over 8}\;+\;D[/tex]

[Janhaa]

diff 1) xy'-5y=2x^5
diff 2) x dy/dx=cos^2x

Integral 1) sin^7x cosx dx
Integral 2) te^3t dt

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2)

[tex]I\;=\;\int {te^{3t}dt\;=\;[/tex][tex]{t\over 3}\cdot e^{3t}[/tex][tex]\;-\;[/tex][tex]{1\over 3}\int {e^{3t}dt[/tex]

[tex]I\;=\;\int {te^{3t}dt\;=\;[/tex][tex]{t\over 3}\cdot e^{3t}[/tex][tex]\;-\;[/tex][tex]{1\over 9}\cdot e^{3t}\;+\;C[/tex]