1.8E: Exercises for Section 1.8

Exercise \(\PageIndex{1}\)

Solve \( y' + y(x^2 - 1) + xy^6 = 0 \), with \( y(1) = 1\).

Hint: \( n=6 \); multiply with \v= ( y^{1-6} = y^{-5}\)

Answer

\(y=y(x)= \left[ e^{5x^3/3-5x} \left( e^{10/3}+\int_{1}^{x}5t\,e^{-5t^3/3+5t}\,dt \right) \right]^{-1/5} \)

Exercise \(\PageIndex{2}\)

Solve \( 2yy' + 1 = y^2 + x \), with \( y(0) = 1 \).

Answer

\(y= \sqrt{e^x - x} \)

Exercise \(\PageIndex{3}\)

Solve \( y' + xy = y^4 \), with \( y(0) = 1 \).

Answer

\(y= \left[ e^{3x^2/2} \left( 1 - 3\int_0^x e^{-3t^2/2}\,dt \right) \right]^{-1/3} \)

Exercise \(\PageIndex{4}\)

Solve \( yy' + x = \sqrt {x^2 + y^2} \).

Answer

\( \sqrt{x^2 + y^2} = x + C \)

Exercise \(\PageIndex{5}\)

Solve \( y' = {(x +y -1)}^2 \).

Answer

\(y= 1 - x + \tan(x + C) \)

Exercise \(\PageIndex{6}\)

Solve \( y' = \dfrac {x^2 - y^2}{xy} \), with \( y(1) = 2 \).

Hint: Multiply with \( y \) and set \( v = y^2 \)

Answer

\(y= \sqrt{\frac{x^2}{2}+\frac{7}{2x^2}}\)

Exercise \(\PageIndex{7}\)

Solve \(xy' + y + y^2 = 0\), \(y(1) = 2\).

Answer

\(y=\frac{2}{3x-2}\)

Exercise \(\PageIndex{8}\)

Solve \( xy' + y + x = 0 \), \(y(1) = 1\).

Answer

\(y=\frac{3-x^{2}}{2x}\)