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Exercise

$\frac{dy}{dt}=2x+sin\left(t\right)+2cos\left(t\right)$

Step-by-step Solution

1

Multiply both sides of the equation by $dt$

$dy=\left(2x+\sin\left(t\right)+2\cos\left(t\right)\right)dt$
2

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

$\int1dy=\int\left(2x+\sin\left(t\right)+2\cos\left(t\right)\right)dt$
3

Expand the integral $\int\left(2x+\sin\left(t\right)+2\cos\left(t\right)\right)dt$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately

$\int1dy=\int2xdt+\int\sin\left(t\right)dt+\int2\cos\left(t\right)dt$
4

Solve the integral $\int1dy$ and replace the result in the differential equation

$y=\int2xdt+\int\sin\left(t\right)dt+\int2\cos\left(t\right)dt$
5

Solve the integral $\int2xdt+\int\sin\left(t\right)dt+\int2\cos\left(t\right)dt$ and replace the result in the differential equation

$y=2xt-\cos\left(t\right)+2\sin\left(t\right)+C_0$

Final answer to the exercise

$y=2xt-\cos\left(t\right)+2\sin\left(t\right)+C_0$

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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