Pergunta
Solve the differential equation
y=3e(x3)C+e(x3),C∈R
Evaluate
dxdy+3x2y=x2
Move the expression to the right side
dxdy=x2−3x2y
Rewrite the expression
dxdy=x2(1−3y)
Rewrite the expression
1−3y1×dxdy=x2(1−3y)×1−3y1
Multiply the terms
1−3y1×dxdy=x2
Transform the expression
1−3y1×dy=x2dx
Integrate the left-hand side of the equation with respect to y and the right-hand side of the equation with respect to x
∫1−3y1dy=∫x2dx
Calculate
Mais Passos
Evaluate
∫1−3y1dy
Rewrite the expression
∫−31×−31+y1dy
Use the property of integral ∫kf(x)dx=k∫f(x)dx
−31×∫−31+y1dy
Use the property of integral ∫ax+b1dx=a1ln(ax+b)
−31ln(y−31)
Add the constant of integral C1
−31ln(y−31)+C1,C1∈R
−31ln(y−31)+C1=∫x2dx,C1∈R
Calculate
Mais Passos
Evaluate
∫x2dx
Use the property of integral ∫xndx=n+1xn+1
2+1x2+1
Add the numbers
2+1x3
Add the numbers
3x3
Add the constant of integral C2
3x3+C2,C2∈R
−31ln(y−31)+C1=3x3+C2,C1∈R,C2∈R
Since the integral constants C1 and C2 are arbitrary constants, replace them with constant C
−31ln(y−31)=3x3+C,C∈R
Calculate
Mais Passos
Evaluate
−31ln(y−31)=3x3+C
Change the sign
31ln(y−31)=−3x3+C
Multiply by the reciprocal
31ln(y−31)×3=(−3x3+C)×3
Multiply
ln(y−31)=(−3x3+C)×3
Multiply
Mais Passos
Evaluate
(−3x3+C)×3
Apply the distributive property
−3x3×3+C×3
Multiply the terms
−x3+C×3
Since C is a constant,replace the C×3 with the constant C
−x3+C
ln(y−31)=−x3+C
Convert the logarithm into exponential form using the fact that logax=b is equal to x=ab
y−31=e−x3+C
Move the constant to the right-hand side and change its sign
y=e−x3+C+31
Calculate
y=33e−x3+C+1
y=33e−x3+C+1,C∈R
Rewrite the expression
Mais Passos
Evaluate
e−x3+C
Use am+n=am×an to expand the expression
eC×e−x3
Since the expression eC is a constant,it is possible to denote that whole expression as a constant C
Ce−x3
y=33Ce−x3+1,C∈R
Solução
y=3e(x3)C+e(x3),C∈R
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