Question

Solve differential equation 2xy-9x^2+(2y+x^2+1)dy/dx=0, y(0)= -3

First order differential equations
ANSWERED
asked 2020-12-29
Solve differential equation \(2xy-9x^2+(2y+x^2+1)dy/dx=0\), y(0)= -3

Answers (1)

2020-12-30

\((2xy-9x^2)dx+(2y+x^2+1)dy=0\)
Comparing with M dx + N dy = 0 we get
\(M= 2xy-9x^2\), \(N= 2y+x^2+1\)
\((dM)/dy= 2x-0= 2x\)
\((dN)/dx= 0+2x+0= 2x\)
\((dM)/dy= (dN)/dx\)
int Mdx+int (terms of N not containing x) dy=C
\(=> \int(2xy-9x^2)dx+\int(2y+1)dy=C\)
\(=> (2y x^2/2)−9 x^3/3+2 y^2/2+y=C\)
\(\int x^2y−3x^3+y^2+y=C\)
\(\int y^2+(x^2+1)y−3x^3=C\) (1)
Given y(0) = −3
substitute x=0 and y= -3 in (1) we get
\((−3)^2+(0^2+1)(−3)−3(0^3)=C\)
9−3=C int C=6
substitute C=6 in (1) we get
\(y^2+(x^2+1)y−3x^3=6\)
\(\int y^2+(x^2+1)y−3x^3−6=0\)
\(2xy−9x^2+(2y+x^2+1)dy/dx=0\)
\(y(0)= −3 is y^2+(x^2+1)y\)
\(−3x^3−6=0\)

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