Linear
equations
y²dx
+ (x²-xy-y2) dy = 0 solve by integrating factor
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I am in 4 th class so I don't know this question answer
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Use a substitution: ⎧⎩⎨⎪⎪u=yxy=uxdy=udx+xdu
y2dx+(x2−xy−y2)dy=0
(ux)2dx+(x2−x(ux)−(ux)2)(udx+xdu)=0
u2x2dx+x2(1−u−u2)(udx+xdu)=0
u2dx+(1−u−u2)(udx+xdu)=0
(u2+u−u2−u3)dx+x(1−u−u2)du
(u−u3)dx=−x(1−u−u2)du
1xdx=−1−u−u2u−u3
1xdx=12(1u+1−1u−1−2u)du
∫2xdx=∫(1u+1−1u−1−2u)du
2lnx=ln|u+1|−ln|u−1|−2ln|u|+C0
lnx2+lnu2+ln|u−1|−ln|u+1|=C0
lnx2+lny2x2+ln∣∣yx−1∣∣−ln∣∣yx+1∣∣=C0
ln∣∣y2(y−x)y+x∣∣=C0
y2(y−x)y+x=C
Step-by-step explanation:
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