Let us solve the differential equation
dxdy=3x+2y+23x+2y, y(−1)=−1.
Let us use the transformation u=3x+2y. Then dxdu=3+2dxdy, and hence dxdy=21(dxdu−3).
It follows that
21(dxdu−3)=u+2u, and hence dxdu=u+22u+3=u+25u+6.
We get the equation dx=5u+6(u+2)du, and hence ∫dx=∫5u+6(u+2)du=51∫5u+6(5u+10)du=51∫(1+5u+64)du=5u+254ln∣5u+6∣+C.
Therefore,
x=53x+2y+254ln∣5(3x+2y)+6∣+C
Since y(−1)=−1, we conclude that −1=−1+254ln19+C, and thus C=−254ln19.
We conclude that the solution of dxdy=3x+2y+23x+2y, y(−1)=−1 is
x=53x+2y+254ln∣15x+10y+6∣−254ln19.
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