document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184077>Question 1184077</A>: <I><SPAN STYLE=\"word-break: break-all;\">  A Bernoulli differential equation is one of the form<BR>\n" );
document.write( "dy/dx+P(x)y=Q(x)y^n.\r<BR>\n" );
document.write( "\n" );
document.write( "Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=y1−n transforms the Bernoulli equation into the linear equation\r<BR>\n" );
document.write( "\n" );
document.write( "du/dx+(1−n)P(x)u=(1−n)Q(x).\r<BR>\n" );
document.write( "\n" );
document.write( "Use an appropriate substitution to solve the equation\r<BR>\n" );
document.write( "\n" );
document.write( "y′−(7/x)y=(y^4/x^5), and find the solution that satisfies y(1)=1.\r<BR>\n" );
document.write( "\n" );
document.write( "y(x)=  ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #815045 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=815045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+y%5E%281-4%29+=+y%5E%28-3%29 ALIGN=MIDDLE  ALT=\"u+=+y%5E%281-4%29+=+y%5E%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Substituting this into the original D.E. and simplifying, you get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22u%27%22+%2B+%2821%2Fx%29u+=+-3%2Fx%5E5 ALIGN=MIDDLE  ALT=\"%22u%27%22+%2B+%2821%2Fx%29u+=+-3%2Fx%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The integrating factor is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%2821int%28dx%2Fx%29%29+=+x%5E21 ALIGN=MIDDLE  ALT=\"e%5E%2821int%28dx%2Fx%29%29+=+x%5E21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "After multiplying D.E. with the integrating factor and integrating, \r<BR>\n" );
document.write( "\n" );
document.write( "you get \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ux%5E21+=+%28-3%2F17%29x%5E17+%2B+C ALIGN=MIDDLE  ALT=\"ux%5E21+=+%28-3%2F17%29x%5E17+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%28-3%29x%5E21+=+%28-3%2F17%29x%5E17+%2B+C ALIGN=MIDDLE  ALT=\"y%5E%28-3%29x%5E21+=+%28-3%2F17%29x%5E17+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy%5E3+=+-3%2F%2817x%5E4%29+%2B+C%2Fx%5E21 ALIGN=MIDDLE  ALT=\"1%2Fy%5E3+=+-3%2F%2817x%5E4%29+%2B+C%2Fx%5E21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solving for C when y(1) = 1 gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+20%2F17 ALIGN=MIDDLE  ALT=\"C+=+20%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so that\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy%5E3+=+-3%2F%2817x%5E4%29+%2B+20%2F%2817x%5E21%29 ALIGN=MIDDLE  ALT=\"1%2Fy%5E3+=+-3%2F%2817x%5E4%29+%2B+20%2F%2817x%5E21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );