Algebra
Calculus
Trigonometry
Matrix
Question
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
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
More Steps
Evaluate
∫1−3y1dy
Use ∫ax+b1dx=a1lnax+b to evaluate the integral
−31ln(3y−1)
Add the constant of integral C1
−31ln(3y−1)+C1,C1∈R
−31ln(3y−1)+C1=∫x2dx,C1∈R
Calculate
More Steps
Evaluate
∫x2dx
Use ∫xndx=n+1xn+1,n=−1 to evaluate the integral
3x3
Simplify
31x3
Add the constant of integral C2
31x3+C2,C2∈R
−31ln(3y−1)+C1=31x3+C2,C1∈R,C2∈R
Since the integral constants C1 and C2 are arbitrary constants, replace them with constant C
−31ln(3y−1)=31x3+C,C∈R
Calculate
More Steps
Evaluate
−31ln(3y−1)=31x3+C
Calculate the product
−31ln(3y−1)=3x3+C
Divide both sides
ln(3y−1)=−313x3+C
Calculate
More Steps
Evaluate
−313x3+C
Rewrite the expression
−313x3+C
Multiply by the reciprocal
3x3+C×(−3)
Multiplying or dividing an odd number of negative terms equals a negative
−3x3+C×3
Reduce the numbers
−(x3+C)
Calculate
−x3+C
ln(3y−1)=−x3+C
Convert the logarithm into exponential form using the fact that logax=b is equal to x=ab
3y−1=e−x3+C
Move the constant to the right-hand side and change its sign
3y=e−x3+C+1
Multiply both sides of the equation by 31
3y×31=(e−x3+C+1)×31
Calculate the product
3y×31=3e−x3+C+1
Cancel out the greatest common factor 3
y=3e−x3+C+1
y=3e−x3+C+1,C∈R
Rewrite the expression
More Steps
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=3Ce−x3+1,C∈R
Solution
y=3e(x3)C+e(x3),C∈R
Show Solution
